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Unformatted text preview: 143 4.1 3! Formal Definition of the Derivative Figure 4.19 The function f(x) = x”3 has a vertical
tangent line at x = 0. It is therefore not differentiable at
x = 0. Section 4.1 Problems 1 _ In Problems 1—8, ﬁnd the derivative at the indicated point from the
graph of each function. 1. f(x)=5:x=1
3. f(x)=4x—3;x=—1
5. f(x)=2x2;x=0 7. f(x) = cosx;x =0 2. f(x)=—3x;x=—2
4. f(x)=5x+l;x=0
6 f(x)=(x+2)2;x=l 8. f(x) = sinx;x z; In Problems 9—16, ﬁnd 6 so that f ’(c) = 0. 9. f(x): —3x2+l 10. f(x)=~x2+4 ll. f(x) = (x  2)2 12. f(x) = (x +3)2 13. f(x)=x2—6x+9 14. f(x)=x2+4x+4. 15. f(x) = sin In Problems 1720, compute f (c + h) — f (c) at the indicated point.
17. f(x) = 2x + 1; c = 2 18. f(x) = 3x2: c =1 19. f(x)=./§;c=4 20. f(x)=%;c=—2 21. (it) Use the formal deﬁnition of the derivative to ﬁnd the
derivative ofy = 5x2 at x = —1. (b) Show that the point (—1,5) is on the graph of y = 5x2, and
ﬁnd the equation of the tangent line at the point (—1. 5). (C) Graph )2 = 5x2 and the tangent line at the point (1. 5) in the
same coordinate system. 22. (2) Use the formal deﬁnition to ﬁnd the derivative of y z:
~2x2 at x = 1. (b) Show that the point (1. —2) is on the graph of y = —2x2. and
ﬁnd the equation of the tangent line at the point (1, —2). (C) Graph y = ——2x2 and the tangent line at the point (1, —2) in
the same coordinate system. 23. (a) Use the formal deﬁnition to ﬁnd the derivative of y :=
1 — x3 at x = 2. (b) Show that the point (2, —7) is on the graph of y = 1—x3, and
ﬁnd the equation of the normal line at the point (2. —7). (t‘) Graph y = 1 — x3 and the tangent line at the point (2. —7) in
the same coordinate system. 24 (8) Use the formal deﬁnition to ﬁnd the derivative of y = in x = 2. l6. cos(rr — x) ’2 ' x’ the equation of the normal line at the point (2. %). (c) Graph y = J1: and the tangent line at the point (2, %) in the
same coordinate system. 25. Use the formal deﬁnition to ﬁnd the derivative of y=~/7c for x > 0.
26. Use the formal deﬁnition to ﬁnd the derivative of 1 f(x)=x+1 forx gé —1. 27. Find the equation of the tangent line to the curve y = 3x2 at
the point (1, 3). 28. Find the equation of the tangent line to the curve y = 2/x at
the point (2. 1). 29. Find the equation of the tangent line to the curve y = J)? at
the point (4, 2). 30. Find the equation of the tangent line to the curve y = x2 
3x +1 at the point (2, —1). 31. Find the equation of the normal line to the curve y = —3x2
at the point (—1, —3). 32. Find the equation of the normal line to the curve y = 4/x at g
the point (—1, —4). l 33. Find the equation of the normal line to the curVe y = 2x2 — 1
at the point (1, 1). 34. Find the equation of the normal line to the curve y = ‘/x  1
at the point (5, 2). 35. The following limit represents the derivative of a function f .l
at the point (a, f(a)):  _ 2(a + h)2 — 202 hm ———'—
h—rO h Find f(x). t 144 Chapter4 l Differentiation 36. The following limit represents the derivative of a function f
at the point (a, f ((1)): . 4(a + h)3 — 403
lim ———
h—DD h Find f (x).
37. The following limit represents the derivative of a function f
at the point (a‘. f (a)):
1 1
. (2%)!“ '3
hm A40 Find f and a. 38. The following limit represents the derivative of a function f
at the point (a, f (a)): " " ' ' ‘ sin(% + h) — sin 1'6 Find f and a. 39. Velocity A car moves along a straight road. Its location at
time t is given by s(t) =20t2, 0 5t 5 2 where t is measured in hours and 30) is measured in kilometers.
(3) Graph 30) for 0 s t 5 2. (b) Find the average velocity of the car between t = O and t = 2.
Illustrate the average velocity on the graph of s(t). (c) Use calculus to ﬁnd the instantaneous velocity of the car at
t = 1. Illustrate the instantaneous velocity on the graph of s(t). 40. Velocity A train moves along a straight line. Its location at
time t is given by 100
30) = T: 15:55
where t is measured in hours and s(r) is measured in kilometers.‘
(a) Graph 50) for 1 s t s S. (b) Find the average velocity of the train between t = 1 and r = 5. Where on the graph of :0) can you ﬁnd the average
velocity? (c) Use calculus to find the instantaneous velocity of the train at
t = 2. Where on the graph of s(t) can you ﬁnd the instantaneous
velocity? What is the speed of the train at t = 2? 41. Velocity If :0) denotes the position of an object that moves
along a straight line, then As/At. called the averagevelocity. is
the average rate of change of s(t), and v(t) = ds/dt, called the
(instantaneous) velocity, is the instantaneous rate of change of
so). The speed of the object is the absolute value of the velocity,
lv(t)l Suppose now that a car moves along a straight road. The
location at time t is given by 160 s(t)=—3—t2. 05:51 where t is measured in hours and 30) is measured in kilometers
(:1) Where is the car at t = 3/4, and where is it at t = 1? (h) Find the average velocity of the car between t = 3/4 and
t = 1. (c) Find the velocity and the speed of the car at t = 3/4. 42. Velocity Suppose a particle moves along a straight line. The
position at time t is given by ‘ s(r)=3t—22. tzo where t is measured in seconds and s(r) is measured in meters.
(2) Graph s(t) fort z 0. (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time 0? (ii) Is there another time at which the particle visits the location
where it was at time 0? (ill) How far' to the right on the straight line does the particle
travel? (iv) How far to the left on the straight line does the particle
travel?" " ” " ‘ ' ' (v) Where is the velocity positive? where negative? equal to 0?
(c) Find the velocity of the particle. (d) When is the velocity of the particle equal to 1 m/s? 43. 'Iilman’s Resource Model In Subsection 4.1.2, we considered
Tilman’s resource model. Denote the biomass at time t b B t) assume 8 ldB
§E=f(R)m where R denotes the resource level, R
f(R) = 200m and m = 40. Use the graphical approach to ﬁnd the value R‘ at
which 71,114 = 0. Then compute R' by solving %% = 0.
44. Exponential Growth Assume that N (t) denotes the size of
a population at time t and that N(r) satisﬁes the differential
equation dN _ r dt _
where r is a constant. (it) Find the per capita growth rate.
(b) Assume that r < 0 and that N (0) = 20. Is the population size  at time 1 greater than 20 or less than 20? Explain your answer. 45. Logistic Growth Assume that 'N(t) denotes the size of a population at time t and that N(t) satisﬁes the differential
equation  Let f(N) = 3N(1 — %) for N z 0. Graph f(N) as a function of
N and identify all equilibria (i.e., all points where % = 0). I 46. Island Model Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies
both the mainland and the islands, but, although it is present on
the mainland at all times, it frequently goes extinct on the islands.
Islands can be recolonized by migrants from the mainland. The
following model keeps track of the fraction of islands occupied:
Denote the fraction of islands occupied at time t by p(!). Assume
that each island experiences a constant risk of extinction and that vacant islands (the fraction 1— p) are colonized from the mainland
at a constant rate. Then dp
I—c(1—p)—ep where c and e are positive constants. (a) The gain from colonization is f(p) = c(1 — p) and the
i055 from extinction is gtp) l= ep. Graph f ( pi and glp) for
0 < p S i in the same coordinate system. Explain why the two a— hs intersect whenevere and c are both posmve. Compute the
ibiiit of intersection and interpret its biological meaning. (b) The parameter c measures how quickly a vacant island
becomes colonized from the mainland. The closer the islands, the
larger is the value of c. Use your graph in (a) to explain what
happens to the pomt of intersection of the two lines as c increases.
interpret your result in biological terms. 47. Chemical Reaction Consider the chemical reaction
A + B ~+ AB If,r(!) denotes the concentration of AB at time I, then i"; = k(a — x)(b — x) i where k is a positive constant and a and b denote the
concentrations of A and B, respectively, at time 0. Assume that
k = 3, a = 7, and b = 4. For what values of x is dx/tlt = 0'? 48. Chemical Reaction Consider the autocatalytic reaction
A + X —> X which was introduced in Problem 30 of Section 1.2. Find a
differential equation that describes the rate of change of the
concentration of the product X. 49. Logistic Growth Suppose that the rate of change of the size
ofa population is given by where N = N(t) denotes the size of the population at time I and
r and K are positive constants. Find the equilibrium size of the
population—that is, the size at which the rate of change is equal
to 0. Use your answer to explain why K is called the carrying
capacity. 50. Biotic Diversity (Adapted from Valentine, 1985.) Walker and
Valentine ([984) suggested a model for species diversity which
assumes that species extinction rates are independent of diversity
but speciation rates are regulated by competition. Denoting the
number of species at time I by N(t), the speciation rate by b. and
the extinction rate by a, they used the model (IN N ~ = N b " _ ' 1]! i: (1 K) a]
whefc K denotes the number of “niches.” or potential Places for
speCies in the ecosystem. i“) Irind possible eqiiilibria under the condition a < b. I 4.2 The Power Rule, the Basic Rules
of Polynomials * 4.2 *1 Basic Rules 145 (b) Use your result in (a) to explain the following statement by
Valentine (.1985): In this situation, ecosystems are never “full,” with all potential niches occupied by species so long as the
extinction rate is above zero. (c) What happens when a _>_ b? 1 4.1.3 51. Which of the following statements is true? (A) If f (x) is continuous, then f (x) is differentiable. (B) If f(x) is differentiable, then f(x) is continuous. 52. Explain the relationship between continuity and differentia
bility. 53. Sketch the graph of a function that is continuous at all points
in its domain and differentiable in the domain except at one point. 54. Sketch the graph of a periodic function defined on R that is
continuous at all points in its domain and differentiable in the domain except at c = k, k e Z.
55. lt'f(x) is differentiable for allx e R except atx = c. is it true
» '  ' ruranswer. In Problems 56—69, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the
largest possible domain.) 56. _v = Ix —2] 57. y = —lx +5l
58.y=2x3 59.y=lx+2l~l
l 1
60. = 61. =
y 2+x y x—
3—x x—l
62. = 63. =
y 3+x y x+1
64. y = ixz 31 65. y = 12x2 — 1
x forx<0
66. = _
ﬁx) ix+l t‘orx>0
2i: forx<1
67. = —'
fa) [x+2 forx>1
x2 for.t<—l
68 '. = “
I“) 'Z—xz forx>—l
2
69.f(x)=[x +1 forng
e" torx>0 70. Suppose the function f(x) is piecewise defined; that is,
f(x) = f1(x)forx 5 a and f(.i) = f;(x) forx > (1. Assume that
f,(.r) is continuous and differentiable forx < a and that f2(x) is
continuous and differentiable for .t‘ > (1. Sketch graphs of f(x)
for the following three cases: (9) f(x) is continuous and differentiable at x = a.
(b) fix) is continuous, but not differentiable. atx = a.
(c) f(x) is neither continuous nor differentiable at .t = a. of Differentiation, and the Derivatives In this section, we will begin a systematic treatment of the computation of derivatives.
Knowing how to differentiate is fundamental to your understanding of the rest of the
course. Although computer software is now available to compute derivatives of many functions (such as y = (1):” or _v = 3““). it is nonetheless important that you master
the techniques of differentiation. 4.2 I Basic Rules 149 XAMP54 V Tangent and hiorttial Lines If f (x) = 2x3 — 3x +1, ﬁnd the tangent and normal lines
at (—1, 2). Solution The slope of the tangent line at (—1. 2) is f ’(—1). We begin calculating this derivative as follows:
f’(x) = 6x2 — 3 Evaluating f ’(x) at x = —1, we get
f’(—1) = 6(—1)2 — 3 = 3
Therefore, the equation of the tangent line at (—1, 2) is
y"2=3(x(l)). or y=3x+5 To ﬁnd the equation of the normal line, recall that the normal, line is
perpendicular to the tangent line; hence, the slope m of the normal line is given by 1 1 ~f’(—1) = 3 The normal line goes through the point (—1, 2) as well. The equation of the normal 1
y—2=~§(X(—1)), 0r Y——§x+§ The graph of f (x), including the tangent and normal lines at (—1, 2), is shown in
Figure 4.22. l Tangent line Normal line Figure 4.22 The graph of f(x) = 2x3 — 3x +1,
together with the tangent and normal lines at (—1. 2). Look again at the last example: When we computed f ’(—1), we ﬁrst computed
f ’(x); the second step was to evaluate f ’(x) at x = —1. It makes no sense to plug
—1 into f (x) and then differentiate the result. Since f (—1) = 2 is a constant, the
derivative would be 0, which is obviously not f’(—1). Just look at Figure 4.22 to convince yourself. The notation f ’(—1) means that we evaluate the function f ’ (x) at
x = —1. Section 4.2 Problems Differentiate the functions given in Problems [—22 with respect to 7. g(s) 537 + 233 — 55 8. g(s) = 3 — 4s2 — 453
the independent variable. 1 " 1
1 f(x)=4x3—7x+1 2. f(x) = ~3x4+5x2 9 h(t)=—§r‘+4t 10. W) = 5t2—3t+2
3. f(x) = —2x5 + 7x —4 4. f(x) = ~3x‘ + 6x2 — 2 ‘
. 11. f(r) =xzsin£+tan£ 12 f(x) =2x3cos£+cos£
s. f(x)=3—4x—5x2 6. j(x)=—1+3x2—2x‘  3 4 ' 3 6 150 Chapter4 I Differentiation 71' TI . x = —3x‘ tan — — cot — 35. Differentiate
13 ﬂ ) 6 73 W!) = Vo(1 + yt)
14. f (x ) = x2 560% + 3x 866 I with respect to t. Assume that V0 and y are positive constants.
1 36. Differentiate NkT
 — __ _ 2 3 _ 4
15.f(t)=13e2+t+e1 16. f(x)_er x pa): V
17. f (s) = sae3 + 3c 18. f (x) = i + ezx + e with respect to T. Assume that N , k, and V are positive constants.
‘ :3 x4 2 37. Differentiate
__ 3 _ a a = _ _ _ __ N
19. f(x) —20x 4x +9x 20. f(x) 15 20 + 15 gm) ___ N (I _ E)
2
2], f(x) = 7H3 —— .1— + it 22. f (x) = 7rer — x—1 with respect to N. Assume that K is a positive constant.
23 Differentiate 7' n e 38. Differentiate
. __ 3
f“) T “x M) = rt! (1 — —)
with respect to x. Assume that a is a constant. K
24. Differentiate 3 with respect to N. Assume that K and r are positive constants
f (x) = x + a 39. Differentiate
with respect to x. Assume that a is a constant. N
25. Differentiate g(N) = er (1 — f (x) = ax2 — 2a with respect to 1;. Assume t at a 15 a constant. ‘ _
26. Differentiate 40. Differentiate f(x) = a2):4 — 2m:2
with respect to x. Assume that a is a constant.
27. Differentiate N
g(N) = rN (a — N) (1 — —)
K
with respect to N. Assume that r, a, and K are positive constants. _ z —
MS) — "S r 41. Differentiate with respect to s. Assume that r is a constant. RH) 23,5 k4 4
28. Differentiate — 15 czhs
._ 2 . . .
f(r)  ’5 — 7 With respect to T. Assume that k, c. and h are posrtive constants. With respec‘ to r‘ Assume mm 5 is a consmnt' In Problems 42—48, ﬁnd the tangent line, in standard form, to y = 29. Differentiate f (x) at the indicated point.
42. y: 3x2 4x +7,atx =2 23
x =rsx —rx+s
ﬂ) 43.y=7x3+2x—1,atx=—3 with respect to x. Assume that r and s are constants. 44. y = —2x3  3x + 1, at x = 1
30. Differentiate 45. y = 2x‘ — 5x, at x = 1
r+x 46.y=—x3—2x2,atx=0
f(x): rs, —rsx+(r+s)x—rs 47_y=_j_ix2_ﬁ,atx=4
with respect to x. Assume that r and s are nonzero constants. 5 n 3
31. Differentiate 48' y = 37” _ Ex ’ a” = 1
N2 In Problems 49—54, ﬁnd the normal line, in standard form, to y =
f (N ) = (b — 1)N i  T f (x) at the indicated point. 49. y =2+x2,atx = —1 with respect to N. Assume that b is a nonzero constant. 50. y = 1 _ 3x2 at x = __2
32. Differentiate 51. y = ﬁx‘ — Zﬂxz at x = —/5
2 I
f(N)=b_N__+_N. 52.y=—2x2——x,atx=0
K+b 53.y=x33,atx=1 with respect to N. Assume that b and K are positive constants 54. =1—7rx2,atx = 1
33. Differentiate 55. Find thet tiin to
80) = “at _ ("3 angen e __ 2 with respect to t. Assume that a is a constant. f (x)  ax 34. Differentiate at x = 1. Assume that a is a positive constant.
2 56. Find the tangent line to _ 42_ 4 L
h(s)—as as+ai f(x)=ax3—2ax with respect to s. Assume that a is a positive constant. at x = 1. Assume that a is a positive constant. 4.3 I Product Rule and Quotient Rule 151 57. Find the tangent mm to N 73. Find a point on the curve ax2
= _ = 2x2 _
f (X) a. +2 y 1
2 ,
at x = 2. Assume that a is a positive constant. whose tangent line is parallel to the line y = x. 15 there more than
58. Find the tangent line to one such point? If so, ﬁnd all other points with this property.
x2 74. Find a point on the curve
a + 1 at x = a, Assume that a is a positive constant.
59. Find the normal line to f(x) = ax” f(x)=
y =l—3.\:J whose tangent line is parallel to the line y = —x. is there more
than one such point? If so, ﬁnd all other points with this property. 75. Find a point on the curve
at x = ~1. Assume that a is a positive constant. ' = x3 + 2x + 2 , ,
60. Find the normal line to whose tangent line is parallel to the line 3x — y = 2. Is there more
than one such point? If so. ﬁnd all other points with this property. 76. Find a point on the curve f(x) = outz  Box at x = 2. Assume that a is a positive constant.
61. Find the normal line to “2 y=2x3—4x+1 0 whose tangent line is parallel to the line y —2x = 1. Is there more than one such point? If so, ﬁnd all other points with this property.
77. Show that the tangent line to the curve at x = 2. Assume that a is a positive constant.
62. Find the normal line to 3 2 x y=x a+1 at x = 2a. Assume that a is a positive constant. f (x) =
at the point (1. 1) passes through the point (0, ——1). 78. Find all tangent lines to the curve
In Problems 63—70, ﬁnd the coordinates of all of the points of the graph of y = f (x) that have horizontal tangents. y = X2 63. f(x)=.vt2 64. f(x)=2—x2 65. f(x) = 3x — x2 66. f(x) = 4x + 2x2 67. f(x) = 3x3 — x2 68. f(x) = —4x‘ + x3
1 7 3 69. f(x) = ix“  3x3  2x2 70. f(x) = 3x5 — 5x4 y = x2 71. Find a point on the curve that pass through the point (0. —1).
79. Find all tangent lines to the curve that pass through the point (0. —a2), where a is a positive number. y = 4 __ xz 80. How many tangent lines to the curve whose tangent line is parallel to the line y = 2. Is there more than
one such point? If so. ﬁnd all other points with this property. 72. Find a point on the curve
2 81. Suppose that P(x) is a polynomial of degree 4. Is P’(x) a
y = (4 " x) polynomial as well? If yes, what is its degree? Whose tangent line is parallel to the line y' = —3. Is there more 82. Suppose that P(x) is a polynomial of degree k. ls P’(x) a
than one such point? If so, ﬁnd all other points with this property. polynomial as well? If yes, what is its degree? y=x2+2x pass through the point (—é, —3)? I 4.3 The Product and Quotient Rules, and the Derivatives of Rational
and Power Functions '1 4.3.1 The Product Rule The derivative of a sum of differentiable functions is the sum of the derivatives of the
functions The rule for products is not so simple, as can be seen from the following r” . 158 Chapter4 l Differentiation Section 4.3 Problems. I 4.3.1 In Problems 1—16, use the product rule to ﬁnd the derivative with
respect to the independent variable.
1 f0?) = 05 + 5)(X2 3)
2. ffx) = (2x3 —1)(3 +2x2)
3. f(x) = (3x‘ — 5)(2x  5x3)
4. f(x) = (3x‘ — x2 +1)(2xz — 5x3)
1
5. f(x) = (5x2 — 1) (2x + 3):?)
6. f(x) = 2(3x2  2x3)(1 — 5x2) 7. ffx) = goal — 1)(x2+1) 8. f(x) = so:2 +2)(4x2 — 5x4) — 3
9. f(x) = (3x — 1)2 10. f(x) = (4  sz)2 . 2x2  3x + 1 2
11. f(x) = 30— 2x)2 ( 4 )  2 — 2 14. h t) = 4(312 — 1)(2t + 1) 15. g(t) = 3(2:2  5:")2 16. h(s) = (4 — 3:2 + 4551
In Problems 17—20, apply the product rule to ﬁnd the tangent line,
in slope—intercept form, of y = f (x) at the speciﬁed point.
17. f(x) = (3x2  2)(x — 1), atx = 1
18. f(x) = (1—2x)(1+2x), atx = 2
19. for) = 4(21‘ + 3x)(4 — 2x2), at x = 1
20. f(x) = (3x3 — 3)(2 — 2x2), at x = o
In Problems 21 24, apply the product rule to ﬁnd the normal line,
in slape—intercept form, of y = f (x) at the speciﬁed point.
21. f(x) = (1  x)(2 —— x2), atx = 2
22. f(x) = (2x +1)(3x2  1), atx =1
23. f(x) = 5(1—2x)(x +1) ~3,atx = 0 24. f(x) = (2 — x)4(3  x) In Problems 25—28, apply the product rule repeatedly to ﬁnd the
derivative ofy = f(x). 25. f(x) = (2x — 1)(3x +4)(1— x) 26. f(x) = (x — 3)(2 — 3x)(5 — x) 27 f(x) = (x  30(2)!2 +1)(1' 132) 28. f(x) = (2x +1)(4 — xz)(1 + xz) 29. Differentiate 12. f(x) = +2 ,atx: —1 [(x) = we 1)(2x — 1) with respect to x. Assume that a is a positive constant.
30. Differentiate f(x) = (a —x)(a +x) with respect to x. Assume that a is a positive constant.
31. Differentiate f(x) = 2a(x2 — a)2 + a with respect to x. Assume that a is a positive constant.
32. Differentiate
3(x — 1)2
2 + a
with respect to x. Assume that a is a positive constant. f(x) = 33. Differentiate g(t) = (at + 1)2
with respect to t. Assume that a is a positive constant.
34. Differentiate h(r) = Jan —a) +a with respect to t. Assume that a is a positive constant.
35. Suppose that f (2) = —4, g(2) = 3. f’(2) = 1, and g’(2) =
—2. Find (fg)'(2) 36. Suppose that f(2) = —4, 3(2) = 3, f’(2) = 1, and g’(2) =
2. Find (1" + gz)’(2)
In Problems 37—40, assume that f (x) is differentiable. Find an
expression for the derivative of y at x = 1, assuming that f (1) = 2
and f’(1) = —1.
37. y = 2x f (x) 39:_ 38. y = 3::2 f (x)
_ xfw ¥ In Problems 41 —44, assume that f (x) and g(x) are differentiable at
x. Find an expression for the derivative of y. 41 y = 3f (10806) 42. y = [f(x) — 3]g(x)
43 y = [f(x) +23(x)lg(X)
44 y = [—2f(x)  3g(x)ig(x) + 53%) 45. Let B (t) denote the biomass at time t with speciﬁc growth rate
g(B). Show that the speciﬁc growth rate at B = 0 is given by the
slope of the tangent line on the graph of the growth rate at B = 0. 46. Let N (t) denote the size of a population at time t. Differentiate
N
N = N 1—
f( ) r ( K) with respect to N . where r and K are positive constants. 47. Let N(t) denote the size of a population at time t.
Differentiate f(N)=r(aN—NZ) (1—%> with respect to N, where r, K, and a are positive constants.
48. Consider the chemical reaction A+B—>AB If x denotes the concentration of AB at time t, then the reaction
rate R(x) is given by R(x) = k(a — x)(b — x)
where k, a, and b are positive constants Differentiate R(x). I 4.3.2 In Problems 4970, differentiate with respect to the independent
variable. 3x — 1 1  4x3
49. = 50. =
f(x) x +1 f(x) 1_x
3x2—2x+1 x‘+2x—1
51. = —————— , = ._.__—
fm 2x+1 52 f“) 5x22x+1
3—):3 1+2);2 4x‘
53. = . = .___._—
ﬁx) 1 —x 54 ﬁx) 3x3 —5x5 r "I Izm 3! +71 M W 7 r 3  t2
= —— 56. ht =
55. Mr) ,H () (Hm
4——2.r2 _ 2s3—4sz+5s——7
57.f(s)=1_s 58f(5)=—(32—:§)—2—~ 60. f(x) = ﬂu" — 5x2) 59 f(x) = .50: — 1) J— 2
5x(1+ x )
61. f(x) = @034) 62, f(x) ._. T 1 1
63f(x) =x3‘F 64.f(x)=x5—; 3 —] 2x2 _ 3
65. f(x) =2x2__ :3 66' “)0 = __x3 + 4x4 s”3 — 1 51/7 _ 52/7
67. g(s) = 52/3 _1 68. g(s) = 33/7 +sm
2
69. f(x) = (1— 2x) (m+ TE)
1 70 for) = (x3 — 3x2 +2) (WM 7; ‘ I) In Problems 71—74, ﬁnd the tangent line, in slope—intercept form,
ofy = f (x) at the specified point. x
—x3+5’
 3 4 + 2 atxl
72. ﬂat) —x ‘5 x2, _
73. f(x) =2x:5,atx=2 74. f(x)=./:€(x3—1),atx=l
x 75. Differentiate x
a
ﬁx) — 3 + x with respect to x. Assume that a is a positive constant.
76. Differentiate ax
f(x) = k +x with respect to x. Assume that a and k are positive constants.
77. Differentiate as:2 for) = 4 +Jt2 with respect to x. Assume that a is a positive constant.
78. Differentiate axz “’0 = k2 + x2 with respect to x. Assume that a and k are positive constants.
79. Differentiate Rn
ﬂm=w+m with respect to R. Assume that k is a positive constant and n is a
positive integer. 80. Differentiate 11(1) = JE(1—a)+a with respect to t. Assume that a is a positive constant. 4.4 I Chain Rule 159 81. Differentiate h(t) = J50  a) + at with respect to t. Assume that a is a positive constant.
82. Suppose that f(2) = —4, g(2) = 3, f’(2) = 1, and g’(2) = —2. Find
1 I
— 2
(f) U 83. Suppose that f(2) = 4. £0) = 3. f(2) = 1, and 8'0) = —2. Find
f I
(a) ‘2) In Problems 84—87, assume that f (x) is differentiable. Find an expression for the derivative of y at x = 2, assuming that f (2) =
~l and f’(2) = 1. f(x) xz+4f(X)
84.y=x2+1 85.y=—W In Problems 88—91, assume that f (x) and g(1) are differentiable at
x. Find an expression for the derivative of y. 88 _2f(x)+1 _ f(x)
'y‘ me ' ‘igixii2 2
90.y= " 9Ly=ﬁf<x>ga> f(X) ~g(x) 92. Assume that f (x) is a differentiable function. Find the
derivative of the reciprocal function g(x) = 1 / f (x) at those points
x where f(x) 9E 0. 93. Find the tangent line to the hyperbola yx = c, where c is a
positive constant. at the point (x,. yl) with x1 > 0. Show that the tangent line intersects the xaxis at a point that does not depend
on C. 94. (Adapted from Raf/f I992) The males in the frog species
Eleutherodactylus coqui (found in Puerto Rico) take care of their
brood. On the other hand, while they protect the eggs, they cannot
ﬁnd other mates and therefore cannot increase their number
of offspring. On the other hand, if they do not spend enough
time with their brood, then the offspring might not survive. The
proportion w(t) of offspring hatching per unit time is given as
a function of (1) the probability f(t) of hatching if time t is spent brooding, and (2) the cost C associated with the time spent
searching for other mates: f (t) w(t)=C+t Find the derivative of w(t). I 4.4 The Chain Rule and Higher Derivatives
l 4.4.1 The Chain Rule In Section 1.2, we deﬁned the composition of functions. To ﬁnd the derivative of
composite functions. we need the chain rule, the proof of which is given at the end of this section. 172 Chapter4 I Differentiation (b) To ﬁnd the time it takes the object to hit the ground, we set s(t) = 30 m and
solve for t: 1 n1 2
30m = 5081);? 60 60
2 2
= — t = — N 2.47
t 9.815’ Or 1/9313 5 I , (We need consider only the positivel‘solution.) The velocity at the time of impact is then
m 60 m
t: = .8  ———— a 4.3—
v() gt (91)52’9'815 2 s I T Section 4.4 Problems I 4.4.1 33. Differentiate
In Problems 1—28. differentiate the ﬁtnctions with respect to the This yields g(T) = 0(To  T)3  b 1. f (x) = (x — 3)2 2. f (x) = (4x + 5)3 with respect to T. Assume that a, b, and T0 are positive constants.
3. f(x) = (1 — 32:2)“ 4. f(x) = (5x2 — 3x)3 34. Suppose that f’(x) = 2x + 1. Find the following:
5 f<x>=¢xz+3 6 f(x>=~/2x+7 (a) j—xf(x2)atx=—1 (b) %f(~/§)atx=4
7. =,/3— 3 8. :45 3 4
f(x) 1 x f(x) x; x 35. Suppose that f’(x) = Find the following:
. = 10. = —— d d
9 f(x) (x3 — 2y f(x) (1 — 5x2)3 (a) d—xf(x2 + 3) (b) and): 1)
_ _ 2 3
11. f (x) = j—ZEJZLI—I 12. f (x) = %——2izi)2— In Problems 36—39, assume that f (x) and g(x) are differentiable
_ — x
d . d f(x) 2
fr _ 1 /——2 _1 36. Find —./T——(x) + g(x). 37. Find —— (—— +1) .
13. f(x) = 2 14. f(x) = ————x—= dx dx g(x)
(X — 1) 2 + x2 + 1 d 1 d 2
: 38. Find f 39. Find 15. f(s) = ./s+./E 16. g(t) = t2+ ./r + 1 dx g(x) dx g(2x) +2x
17 t 3 18 h 232 4 In Problems 40—46, ﬁnd ﬁf by applying the chain rule repeatedly.
. l = — . =
g” (:4) (s) ( 1) 40. y: (./1—2x2+1)2 41. y=(./x3—3x+3x)‘
_ 2 2 2
19. f(r) = (r2 — r)3(r + 3r3)" 20. [1(5) = 7:95.97)? 41 y = (1+ 20: + 3)‘) 43 y = (1 + (341:2 — if)
s s — 2 3
x Z): + 1
21 h<x>=¢‘3x‘ lemma/314* 4‘" y" ‘5' y: 23. f(x)=.7/x2—2x+1 24 f(x)=“/2—4x2 (2x+1)2_x 2
25. g(x) = (3s7 — 7.03/2 26. h(t) = (t4 — 505/2 46. y = —3——3
3 2/5 4 m (3;: +1) ~—x
27. h(t) = (3: + 7) 23. ho) = (4:4 + F) I 4,4,2
29. Differentiate In Problems 47—54, ﬁnd 3;”; by implicit differentiation.
f(4\7)=(ax+1)3 47. x2+y2=4 48.y=x2+3yx
with respect to x. Assume that a is a positive constant. 49. x3" + y3/4 = 1 50. xy — y3 = 1
30. Differentiate 1
51. = 2 . — — 3 =
f(x)=./ax2—2 “5 x“ 52 2x)’ y 4
with respect to x. Assume that a is a positive constant. 53. 1r. = X 54. x = by
31. Differentiate y x xy+1
(N) _ bN In Problems 55~57, ﬁnd the lines that are (a) tangential and (b)
g _ (k + N)2 normal to each curve at the given point.
with respect to N. Assume that b and k are positive constants. 55 X2 + ,v2 = 25. (4. —3) (CirCle)
32. D' ' 2 2
'“e'em'a‘e N 56. a— + {)— = 1, (1. Vi) (ellipse)
N = —————
3‘ ) (k + bN)3 x2 2 . y 5 ,
With respect to N . Assume that b and k are positive constants. 57' '23 _ ‘9" = 1' (2? 4) (hYPcrbOId) 58. Lemniscate (a) The curve with equation y2 = x2  x‘ is shaped like the
numeral eight. Find 5;: at G, WE). (1,) Use a graphing calculator to graph the curve in (a). If the
calculator cannot graph implicit functions, graph the upper and
the lower halves of the curve separately; that is, graph Yr =vx2x‘
y2=—vx2—X4 Choose the viewing rectangle —2 5 x 5 2, —1 5 y 5 1. 59. Astroid . d
(3) Consider the curve with equation it”3 + y”3 = 4. Find Hf at (—1.3/3'). (b) Use a graphing calculator to graph the curve in (a). If the
calculator cannot graph implicit functions, graph the upper and
the lower halves of the curve separately. To get the left half of
the graph. make sure that your calculator evaluates x”3 in the
order (x2)”3. Choose the viewing rectangle —10 5 x 5 10,
—10 5 y s 10. 60. Kamper of Eudoxus
onsr ert e curve with equation y2 = 10x4 — x2. Find 5‘: at
(1, 3). (b) Use a graphing calculator to graph the curve in (a). If the
calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately. Choose the viewing
rectangle —3 5 x s 3, —10 5 y 5 10. I 4.4.3
6]. Assume thatx and y are differentiable functions of r. Find ‘5;
when):2 +y2 =1, Z—f =2forx = %,andy > 0. 62. Assume that x and y are differentiable functions of r. Find wheny2 =x2 —x‘, 2—: =1forx = %,andy > 0. 63. Assume thatx and y are differentiable functions of r. Find whenxzy = land 53—: = 3 forx = 2. 64. Assume that u and v are differentiable functions of r. Find Whenu2+ u3=12, ‘ﬁ =2forv=2,andu > 0. 65. Assume that the side length x and the volume V = x3 of a cube are differentiable functions oft. Express dV/dt in terms of
dx/dt. 66. Assume that the radius r and the area A = In2 of a circle are
differentiable functions of r. Express dA/dt in terms of dr/dt. 67. Assume that the radius r and the surface area S = 47w2 of a sphere are differentiable functions of 2. Express dS/dt in terms of
dr/dt. m. if the height of the water in the tank is h, then the volume of
the water is V = nrzh = (25m2)71h = 257th m2. If we drain the
water at a rate of 250 liters per minute, what is the rate at which the water level inside the tank drops? (Note that 1 cubic meter
contains 1000 liters.) Conical tank at the rate of5 cubic feet per minute (i.e., the tank
Stands with its point facing downward). The tank has a height of6 4.4, I Chain Rule. 173 ft and the radius on top is 3 ft. What is the rate at which the water
level is rising when the water is 2 ft deep? (Note that the volume
of a right circular cone of radius r and height}: is V = §7rr2h,) 71. No people start biking from the same point. One bikes east
at 15 mph, the other south at 18 mph. What is the rate at which
the distance between the two people is changing after 20 minutes
and after 40 minutes? 72. Allometric equations describe the scaling relationship be
tween two measurements, such as skull length verSUS body length.
In vertebrates, we typically ﬁnd that [skull length] or [body length]" forO < a < 1. Express the growth rate of the skull length in terms
of the growth rate of the body length. I 4.4.4 In Problems 73—82, ﬁnd the ﬁrst and the second derivatives of each
frmction. 73. f(x) =x3 — 3x2 +1
75 74. f(x) = (2x2 + 4)3 x 52 +2
1 3
77. g(r) = ,/3r3 +2t 78. f(x) = F +1: —x
2x
79. [(s) = 53/2 —l 80. f(x) = x2 +1
1
81. g(t) = 1—5/2 — t”2 82. f(x) = x3 — F 83 Find the ﬁrst 10 derivatives of y = x5. 84. Find f(")(x) and f("+”(x) of f(x) = x". 85. Find a second~degree polynomial p(x) = as:2 + bx + c with
19(0) = 3. p’(0) = 2. and p”(0) = 6 86. The position at time I of a particle that moves along a straight
line is given by the function s(t). The ﬁrst derivative of s(t) is
called the velocity, denoted by v(t); that is, the velocity is the rate of change of the position. The rate of change of the velocity is
called acceleration, denoted by a(t); that is, £110) = a(t) dt Given that v(t) = s'(r), it follows that
d2
(1—1550) = (10) Find the velocity and the acceleration at time t =
following position functions: (a) W) = t2 — 3: (b) s(t) = ,/12 +1 (c) s(r) = t“ — 2t 87. Neglecting air resistance, the height h (in meters) of an object thrown vertically from the ground with initial velocity no is given
by 1 for the l
h(t) = vot — Egtz where g = 9.81 m/s2 is the earth‘s gravitational constant and r is
the time (in seconds) elapsed since the object was released.
(a) Find the velocity and the acceleration of the object. (b) Find the time when the velocity is equal to 0. In which direction is the object traveling right before this time? in which
direction right after this time? 177 4.5 f! Derivatives ofTrigonometric Functions I “Ohm”. I—58, ﬁnd the derivative with respect to the
n independent variable. I I
l. = ZSiHX " (305.! 2' = Sensr _ ZSlnx
3. f(x) = 35in.\' + Scosx  Zsecx 4 f(x) —sin.t + cosx — 3cscx . 5. f“) 6' f(x) = secx  csc x 7. f”) — NinaY) 8' f(x) = cos(5x) 9. f(x) = 25in(3.\' +1) 10‘ f(x) = _3COS(1 _ 2x)
I]. f(x) = tan(4.\') 12. f(x) = cot(2 _ 3x) ‘3' f“) = 25°C“ +2“ 14. f (x) = —3csc(3 — 5..)
15. f(x) = 35in(.\”3) [(5, f(x) = 2mm. _ 3X)
‘7 ft") = “"103 — 3) 18. f(x) = coszqz _ t) 19' m) = “max: . 20 f(x) = —Sin'(2x3 1)
21 f(X) = 4cosxI ~ 2cos~ x 22' f(x) = F5C05(2“X3)+2cosl(x —4) . 23‘ f‘“ = 4°05!" + Zcosx“ 24. f(x) = —3cos‘(3.r3 _ 4)
25. f(x) = 2tttn(1.t2) 26. f (x) = — cott3xJ — 4x)
27. f(x) = —2tan‘(3x — 1) 28. f(x) = Sinx + sin I i — ~ 2 2 ll tan .t‘ —~ C0t.\' H l ' sin(3t)
' = ./ — . 2. =
31. 34(3) coss cos t 3 g(t) c056!)
sin(2t) + 1 . c0t(2x)
2 = ———— 34. x =
33' 5’“) cos(6l) —1 ﬂ ) tan(4.t)
:  ‘3 — l ‘
35. f(x) = 36. f(x) : smx cosx
37. f(x) = sin(2x  l) cos(3x + l)
38. f(x) = tanx cotx
39. f(x) = tant3xz — l)cot(3x2 + i)
40. f(x) = secx eosx
l
41. f(x) = sinx secx 42. f(x) = O 2 ,
sm x + cos” x
l
43 ‘ = —,—T 44. I = I
f ” tan‘x ~sec‘x Mir) sm(3x)
l 1
45. .‘ = ——————— . , =
g“) sin(3.r2 — l) 46 g“) esc2(5x)
l .
47 six) : 4s. h(x) = cot(3x)csc(3x)
49 h 3 1 m
 (X) = ———~—~— 50. = '
tunt2x) — .t‘ gm (sin (2)
51. mo = sin'xx +cos~’s 52. f(x) = (2x3 —x) cosil —x2)
. 2x) 1+ cost3x)
53. i. ) = “M , = m
f r 1 +12 54 f(x) 2x3 _x
I
55 f(x) = t; — . =
til x 56 f(x) sec 1 +x2
5,. f(x) : sec‘x 58. f(x) = esc(3 — x2)
sec‘x ' 1 —x3 59 Find the points on the curve y = sin(%.‘:) that have a
horizontal tangent. 60 Find the points on the curve y = cos2 .\‘ that have a horizontal
tangent. 61. Use the identity CUStu + [it = cosa cos/5  sin or sin [3
and (he definition oi the derivative to show that
(I —— cost = — sin .t
t1.\' 62. Use the quotient rule to show that d 2
—— cot .r = — csc x
dx (Him: Write cotx = 3! sin .r ' 63. Use the quotient rule to show that d ~— secx = secx tan .t
dx [Him: Write secx = (cosx)".]
64. Use the quotient rule to show that d
—— cscx = — cscx cotx
dx [Himz Write cscx = (sin .t')".]
65. Find the derivative of f(x) = sin «x? +1 in t e I envattve 0 f(x) = cos sz +1 67. Find the derivative of f(x) = sin ‘/ 3xJ + 3x 68. Find the derivative of f(x) = cos ./1 — 4x4 69. Find the derivative of f(x) = sin2(.r2 — 1) 70 Find the derivative of f(x) = coszt2x2 + 3)
71. Find the derivative of f(x) = tan"(3x3 — 3)
72. Find the derivative of f(x) = sec2(2x2 — 2) 73. Suppose that the concentration of nitrogen in a lake exhibits
periodic behavior. That is. if we denote the concentration of
nitrogen at time I by C(t), then we assume that (U) = 2 + sin dc
(Ir (b) Use a graphing calculator to graph both ctr) and '7‘} in the
same coordinate system. (a) Find (c) By inspecting the graph in (h). answer the following questions:
(i) When ctr) reaches a maximum. what is the value of ilr/t/l‘.’
(ii) When (Ir/(l! is positive. is ('(I) increasing or decreasing? (iii) What can you say about ('(I) when (IE/(II : 0‘.’ 4.6 :l Derivatives of Exponential Functions 181 where W0 is the amount of material at time 0 and A is called the radioactive decay
rate. Show that W(t) satisﬁes the differential equation Solution We use the chain rule to ﬁnd the derivative of W(t): iWm = Woe—“EM d! w
WU)
That is,
dW
—— = ——),W t
dt ( ) In words, the rate of decay is proportional to the amount of material left. This
equation should remind you of the subtangent problem; there, we wanted to ﬁnd
a function whose derivative is proportional to the function itself. That is exactly the
situation we have in this example: The derivative of W(t) is proportional to W(t). I j EXAMPLE 8* Exponential Growth Find the per capita growth rate of a population whose sizeN t,
W N0) = N(O)e" where N (0) is the population size at time 0 and r is a constant. Solution We ﬁrst ﬁnd the derivative of N0): Since N(0)e” = N(t), we can write dN
—— = rN t
d! ( )
Thus, the per capita growth rate of an exponentially growing population is constant;
that is,
1 (IN _
N (It — r I .8éction4ﬁ Problems: — .‘ Differentiate the functions in Problems [—52 with respect to the 23. f (x) = sin(e1‘ + x) 24. f(.r) = cos(3x ~ fl")
independent variable. 25. f (x) = exp[x —— sin x] 26. f (x) = exp[x2 — 2cosx]
1. f(x.) = e“ 2. f(x) = {7" 27. gts) = exp[secs2] 28. g(s) = exp[tan s3]
3. fm =4el‘1' 4. f(x) = 3e2'5" 29. f(x) = e”‘""' 30. for) =e"‘°°“
5 = e_z"z+j"_l 6, = 34‘2’1Y+l 31. [‘00 = _3e.r2Hnnx 32' f(x) = Ze—rsectlr)
7 rm = s. m) = 33 for) = 2' 34 ftx) = 3‘
9 f(x) = .rer 10. f(x) 2 hr“ 35 fix) = 2"H 36 f(.\') = 3""—l
11. f(x) = xzer 12. f(x) = (3x2 —1)el\'2 37. ftx) = 5v3'' 38. ftx) = 3W“~“'
\  ~‘_
'3 f on: $3 14. f(.r) = a5 39. _f(.\') =23?!H 4o. f(.r) =31 '
,5. _ , _ . 41. 110) :2“ 42. [1(1) 242"”
f”) ‘ 2+» [6' f(") “ w+w 7w I7_ = esinlln) '8. = eunudv) 43' I  44' flx) = 3
19' ftx) = e‘i”"‘?“" 20 = e“"sll‘2"j’ 45 flx) : 2 ' rl'l 46 = 4' “21‘
21 fix) = sin(e") 22. f(x) = cos(e") 47 ’1“) = 5J7 48 “(U = 5" ("Lb Hm 182 Chapter4 I Differentiation 49. g(x) = 22cm 50. g(r) = 2H“
51. g(r) = 3’”5 52. g(r) = 4"”
Compute the limits in Problems 53—56. e2"  1 . 35" — 1
53. l' 54. hm
hf?) h hOO e’l  1 , 2’I — 1
55. lim ' 56. hm
two J}; II—>0 ’1
57. Find the length of the subtangent to the curve y = 2" at the
point (1.2). 58. Find the length of the subtangent to the curve y = exp[x2] at
the point (2. e“) .
59. Population Growth Suppose that the population size at time
t is N (2) == ez’ ,
(a) What is the population size at time 0?
(b) Show that :20 dN
—— = 2N
d: 60. Population Growth Suppose that the population size at time N0) = Noe", t 2 0 where N0 is a positive constant and r is a real number.
(a) What is the population size at time 0? (b) Show that 7.7”” 61. Bacterial Growth Suppose that a bacterial colony grows in
such a way that at time t the population size is N (t) = N (0)2' where N (0) is the population size at time 0. Find the rate of growth
dN/dt. Express your solution in terms of N (1). Show that the
growth rate of the population is proportional to the population
size. 62. Bacterial Growth Suppose that a bacterial colony grows in
such a way that at time t the population size is N (t) = N (0)2' where N (0) is the population size at time 0. Find the per capita
growth rate.
63. Logistic Growth ~ (3) Find the derivative of the logistic growth curve (see Example
3 in Section 3.3) N (t) = ———K—————
” (not ‘ )e‘" where r and K are positive constants and N (0) is the population
size at time 0. (b) Show that N (t),satisﬁes the equation dN N
._=N1___
d: r ( K) lHint: Use the function N (t) given in (a) for the righthand side,
and simplify until you obtain the derivative of N(l) that you
computed in (a).] (c) Plot the per capita rate of growth 3— 4—” as a function of N . and N I“
note that it decreases with increasing population size. 64. Fish Recruitment Model The following model is used in the
ﬁsheries literature to describe the recruitment of ﬁsh as a function
of the size of the parent stock: if we denote the number of recruits
by R and the size of the parent stock by P, then R(P) = aPe‘“, P z o
where a and 13 are positive constants.
(a) Sketch the graph of the function R(P) when [3 = 1 and a = 2. (b) Differentiate R(P) with respect to P.
(c) Find all the points on the curve that have a horizontal tangent. 65. Von Bertalanffy Growth Model The growth of ﬁsh can be
described by the von Bertalanffy growth function L0!) = Lee  (Loo  Lo)?" where x denotes the age of the ﬁsh and k, Log. and Lo are positive constants.
(a) Set L0 = 1 and Lao = 10. Graph L(x) fork = 1.0 andk = 0.1. (b) Interpret L“, and Lo.
(c) Compare the graphs for k = 0.1 and k = 1.0. According to (d) Show that 51th) = ML... — Lo»
dx That is, dL/dx o< Loo — L. What does this proportionality say
about how the rate of growth changes with age? (e) The constant k is the proportionality constant in (d). What
does the value of I: tell you about how quickly a ﬁsh grows? 66. Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time t (measured in days). Assume
that the radioactive decay rate of the material is 0.2/day. Find the
differential equation for the radioactive decay function W(r). 67. Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time i (measured in days). Assume
that the radioactive decay rate of the material is 4/day. Find the
differential equation for the radioactive decay function W(t). 68. Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time I (measured in days). Assume
that the half—life of the material is 3 days. Find the differential
equation for the radioactive decay function W(t). 69. Radioactive Decay Suppose W(r) denotes the amount of a
radioactive material left after time t (measured in days). Assume that the halflife of the material is 5 days. Find the differential
equation for the radioactive decay function W(t). 70. Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time 1. Assume that W(0) = 15 and
that dW
__ =._ w
d: 2 (t) (a) How much material is left at time t = 2?
(b) What is the halflife of this material? 71. Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time t. Assume that W(0) = 6 and
that dW
7'— : ~3W(t) (a) How much material is left at time t = 4?
(b) What is the halflife of the material? 72, Radioactive Decay Suppose W(t) denotes the amount of a
radioactive material left after time t. Assume that W(0) = 10 and W”) :8. (a) Find the differential equation that describes this situation.
(1,) How much material is left at time t = 5? (c) What is the halflife of the material? 4.7 1 Inverse Functions and Logarithms 183 73. Radioactive Decay Suppose W(t) denotes the amount of a radioactive material left after time 1‘. Assume that W(0) = 5 and
W“) = 2. (a) Find the differential equation that describes this situation.
(b) How much material is left at time t = 3?
(c) What is the halflife of the material? 1 4.7 Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse
Tangent Function Recall that the logarithmic function is the inverse of the exponential function. To ﬁnd the derivative of the logarithmic function, we must therefore learn how to compute
the derivative of an inverse function. I 4.7.1 Derivatives of Inverse Functions We begin with an example (Figure 4.32); Let f (x) = x2, x z 0. We computed the
inverse. function of f in Subsection 1.2.6. First note that f (.r) = x2, x 2 O, is one
to one (use the horizontal line test from Subsection 1.2.6); hence. we can deﬁne its ,
 unc ton. eat 7 that we obtain the graph of the inverse function by reﬂecting y = f (x) about the line
y = X] 1. Writey = f(x): y = x
2. Solve for x: x = ﬂ
3. Interchange x and y: y = ﬂ Since the range of f (x), which is the interval [0, 00), becomes the domain for the
inverse function, it follows that f'1(x) = ﬂ forx z 0
We already know the derivative of J}, namely, 1/(2ﬁ). But we will try to ﬁnd the derivative in a different way that we can generalize to get a formula for ﬁnding the
derivative of any inverse function. Let g(x) = f "(x). Then if og)(x) = f[g(x)] = ($32 = X. x30 Figure 4.32 The function y = x2. x z 0, and its inverse function y = J: 3 0. 192 Chapter4 l Differentiation Proof We set y = 2’ and use logarithmic differentiation to obtain d d
—— l = — l ’
dx lny] dx [an
ldy d
_._ = __ 1
ydx dx[r nx]
ldy l
__ =r_.
‘ydx x
Solving for d y /dx yields
d—y = rly =rlx' = rx"l l
dx x . Section 4.7 Problems: I 4.7.1 16. Let In Problems [—6, ﬁnd the inverse of each function and differentiate
each inverse in two ways: (i) Differentiate the inverse function
direct! and it use 4.12 to rid the derivativeo the inverse. f(x) =x2+tanx, x e (~g. 1. f(x)=./2x+1,xz—g 2. f(x)=./x—1,x21 4. f(x) =3x2+2,x 30 3. f(x) 22x2—1,x 20 2—
s. f(x)=3—2x3,x20 6. f(x)=2" 1 the indicated poim
7. Let f(x)=2x2—2_ x20
Find ﬁf'1(x)lr=o. [Note that f(1) = 0.]
8. Let f(x) = —x3+7' x 20
Find ﬁf"(x),=_l. [Note that f(2) = _1_] 9. Let
f(X)=\/x+1, x 20
Find ﬁf"‘(x)x=2. [Note that f(3) = 2.] 10. Let f(x) = «2+x2, x z 0
Find :7 f“(x)x= ﬂ. [Note that f(1)= ﬁ]
11. Let f(x)=x+e". x ER
Find :7 f”1(x)lx___l. [Note that f(0)=1.]
12. Let
f(x) =x +ln(x +1), x > —1 Find ﬂitlung“. [Note that f(0) = 0.]
13. Let f(x) =x ——sinx, x E R
Find iiilamb". [Note that f(x) = IL]
14. Let f(x) =x —cosx, x e R
Find %f_l(x)lx=_l. [Note that f(0) = —1.] 15. Let
7t 7: f(x) = x2 + tanx. x 6 (~53 Find %f_l(.t')lx=0. [Note that f(0) = 0.] x>1 x2—1' In Problems 7—22, use (4.12) to ﬁnd the derivative of the inverse at dx ,gzIrKH' 16
17. Let f(x) = ln(sinx),0 < x (ii/2. Find ﬁflm a
x = —ln2. 18. Let f(x) = ln(tanx),0 < x < n/z. Find ﬁf‘Nx) atx = L"; 19. Letf(x) =x5+x+1. —1 < x <1.Find d—‘i—f'Wx) atx = 1
20. Let f(x) = r“ +x. Find j—xfKx) atx = 1. 21. Let f(x) = e"2’2 + 2):. Find ﬁf—IQ) at x =1. 22. Denote the inverse of y = sinx, —§ 5 x 5 §, by y
arcsinx, —1 _<_ x s 1. Show that d 1 —— arcsinx = dx ./1—x2' 1<x<1 I 4.7.2 In Problems 23—60, differentiate the functions with respect to th. independent variable. (Note that log denotes the logarithm to bas
10. ) 23. f(x) = ln(x +1)
25. f(x) =1n(1 —2x)
27. f(x) = lnt:2 29. f(x) = 1n(2x3 — x)
31. f(x) = (ln x)2 33. f(x) = (lnxz)2 35. f(x) = ln./x2 + 1 24. = ln(3x + 4)
26. f(x) = ln(4 — 3x)
28. f(x) = ln(1 —x2)
30. f(x)=ln(1— x3)
32. f(x) = (lnx)3 34' f(x) = (ln(1— X2»3 36. f(x)=ln\/2x—1———; x 2x
37. f(x)—1n):+1 38. f(.ic)=lnl+x2
1— x x2 —1
39. = . =
f(x) ln1+2X 40 f(x) ln x3 _1
41. f(x) = exp[x — lnx] 42. g(s) = exp[sZ + his]
43. f(x) = ln(sinx) 44. f(x) = ln(cos(1 —— x))
45. f(x) = ln(tanx2) 46. gm = ln(sin2(3s))
47. f(x) = .t lnx 48. f(x) = lenx2
lnx lnt
49. f(x) .. T 50. h(t) = “HZ 51. [10) = sin(ln(3t)) 52. Ms) = ln(lns) 4.8 I Linear Approximation and Error Propagation 193 53. f(x) = In [x2 ; 37 54. f(x) = 103(2):2 — 1) 62. Assume that f(x) is differentiable with respect to x. Show
55. ftx) = log(1 — x2) 56. f(x) = togox3 — x +1!) that ,
d 1
57. f(x) = log(x’ — 3x) ' 58. f(x) = new tanr2) E l" = ‘ ?
59. f (n) =1083(3 + u“) 60 gm = 10g5(3‘  2) a ' 61, Let f(x) = lnx. We know that f’(x) = We will use this I 4.7.3 fact and the deﬁnition of derivatives to show that In Problems 6374, use logarithmic differentiation to ﬁnd the ﬁrst I n derivative ofthe given functions.
lim 0+) =e 63. ftx) =2r*” 64. for) =<2x)1*
"*°° ” 65. f(x) ; (my 66. f(x) = (lnx)3"
(1.) Use the deﬁnition of the derivative to show that 67. f (x) '= x'“ 68. f (x) = xZ’“
_. l/x _. 3/x
I I 1110+,” 69. f(x)—Jx 70. f(x)—x
f (1) = llm ‘ h 71. y = x" 72. y = (x‘Yr
“’0 73. y = 74. y = (cost‘
(b) Show that (a) implies that 75. Differentiate
ln[lim(1+ h)”"] = 1 e219; — 2)3 = A“
11—.0 y ‘ (x2 + l)(3x3  7) (c) Set h = 5 in (b) and let n —+ 00. Show that this implies that 76. Differentiate lim 1+ :9 y=(x2+5)2’ ntoo ’1 II 4.8 Linear Approximation and Error Propagation Suppose we want to ﬁnd an approximation to ln(l.05) without using a calculator.
The method for solving this problem will be useful in many other applications. Let’s
look at the graph of f (x) = lnx (Figure 4.39). We know that In] — 0, and we
see that 1.05 is quite close to l—so close, in fact, that the curve connecting (1, 0) to
(1.05, In 1.05) is close to a straight line. This suggests that we should approximate the
curve by a straight line~but not just any straight line: to the graph of f (x) = lnx at x = 1 (Figure 4.39). We can ﬁnd the equation of the
tangent line without a calculator. We note that the slope of f (x) = lnx at x = 1 is f’(1) = HR] = 1. This, together with the point (1, 0), allows us to ﬁnd the tangent
line at x = 1: L(Xl = f(1)+f'(1)(x —1)= 0+(1)(x  1) =x ~1 Figure 4.39 The tangent line approximation for In x at
x = 1 to approximate ln(l.05). When x is close to l, the
tangent line and the graph ofy = lnx are close (see inset). #— 198 Chapter4 l Differentiation We require that 100 l A—ﬂ = 10. Hence, «we» Ad 10_ That is, we must measure the stem diameter to within an error of 5.4%.
Using the result of Example 5, We could have found the same error immediately. Since 01' A(d) = call8‘ we get s = 1.84, where s is the exponent deﬁned in Example 5. Using AA
we obtain
d 1 AA 10
—— = — — = —— = 5.4
100 4 100 A 1.84
as before. , I Suppose that you wish to determine the percentage error of f (x) from a
measurement of x, where f (x) = In x, x = 10, and the percentage error for x is equal to 2%. Find the percentage error of f (x). Solution The function f (x) is not a power function, so there is no simple rule. We ﬁnd that 1009!— ~ 100f’(x)Ax f f (I) Since we know 1009f, we multiply'and divide the righthand side by x and
rearrange terms to get f’(x)Ax Ax xf’(x)
100 = 100—
f (x) x f 00
Since f’(x) = l/x, atx = 10 we obtain
Ax xf’(x) (10)(1/10) 2
x f (x) mm in 10 in 10 0 9
Thus, the percentage error of f is approximately 0.9%. I 1 Section 4.8 Problems v . v' In Problems 140, use the formula 7_ sin + 0.02) 8. cos __ 0.01)
f (x) z f (a) + f ’(a)(x  a) 9. ln(1.01) 10. e"1
to approximate the value of the given function. Then compare your 1" P’0blem3 1130; “PPTOXimale f(x) 0‘ a by "13 linear “PPTOXI'
result with the value you get from a calculator. maﬁa"
1. J63; let f(x) = ﬁw = 64, andx = 65 L0" = f“) + “‘1’” ‘a)
2. ﬁ;ietf(x)=ﬁ,a=36,andx=3s 11. f(x)=1+xata=0 12. f(x)=l ata=0
3. 3/ 124 4. (7.9)3 2 x
5' (0_99)25 6_ tanmm) l3. f(x) = 1+x ata =1 14. f(x) = 3 _ 2x ata = 2 7 1 15. f(x)=(1+x)2 ata=0 16. f(x)=(1_x)2 17. f(x) :=ln(1+x)ata =0 18. .f(x)=ln(1+2x)ata=o
19, f(x) = logx ata =1 20. f(x) = log(1+x2) ata = 0
21. f(x)==e‘ata=0 22. f(x)=e2‘ata=0 23, f(x) = e“" ata = 0 24. f(x) = e41 ata = 0 25. f(x) = e“' ata =1 26. f(x) = ez’“rl ata = ~—1/2
27, f(x) = (1 + x)"' at a = 0. (Assume that n is a positive
integer.) 28. f(x) = (1 — x)‘" at a
integer.) 29, f(x)=,/1+x2ata=0
1 114
30. ata=1 31. Population Growth Suppose that the per capita growth rate
of a population is 3%; that is, if N (t) denotes the population size
at time t, then ata=0 0. (Assume that n is a positive Suppose also that the population size at time t = 4 is equalto
l 0 Us . it 
at timet = 4.1.
32. Population Growth Suppose that the per capita growth rate
of a population is 2%; that is. if N (t) denotes the population size
at time t, then .00 6.". 0 I IIII ' ‘ ?'3. ——=0. 2
Nd! 0 Suppose also that the population size at time t = 2 is equal to
50. Use a linear approximation to compute the population size at
time t = 2. l. 33. Plant Biomass Suppose that the speciﬁc growth rate of a
plant is 1%; that is, if 80) denotes the biomass at time t, then Suppose that the biomass at time t = 1 is equal to 5 grams. Use a
linear approximation to compute the biomass at time t = 1.1. 34. Plant Biomass Suppose that a certain plant is grown along
a gradient ranging from nitrogenpoor to nitrogenrich soil.
Experimental data show that the average mass per plant grown in
a soil with a total nitrogen content of 1000 mg nitrogen per kg of
soil is 2.7 g and the rate of change of the average mass per plant at
this nitrogen level is 1.05 x 10'3 g per mg change in total nitrogen
per kg soil. Use a linear approximation to predict the average mass per plant grown in a soil with a total nitrogen content of 1100 mg
nitrogen per kg of soil. In Problems 35—40, a measurement error in x affects the accuracy
of the value f (x). In each case, determine an interval of the form [f(x)  Af. f(X) + Af] that reﬂects the measurement error Ax. In each problem, the
quantities given are f (x) and x = true value of x :t: [Ax]. 35. f(x) = 2x,x =1i0.1 36 f(x)=1— 3.“ = —2 $0.3 37. f(x) = 3x2,x = 2 £01 4.8 l Linear Approximation and Error Propagation 199 38. f(x)=ﬁ,x=10:l:0.5
39. f(x)=e’,x=2i0.2
40. f(x)=sinx,x=—1d:0.05 In Problems 41—44, assume that the measurement of x is accurate
within 2%. In each case, determine the error A f in the calculation
of f and ﬁnd the percentage error 100%. The quantities f (x) and
the true value of x are given. 41. f(x)=4x3.x=1.5 42. f(x)=x‘/‘,x=10 » 1 43. f(x) = lnx,x =20 44. f(x)=1+x,x = 4
45. The volume V of a spherical cell of radius r is given by
V(r) = gar3 If you can determine the radius to within an accuracy of 3%, how
accurate is your calculation of the volume? 46. Polseuille’s Law The speed u of blood ﬂowing along the
central axis of an artery of radius R is given by Poiseuille’s law, v(R) = cR2 where c is a constant. If you can determine the radius of the artery to within an accuracy of 5%, how accurate is your calculation of
the speed? 47. Allometric Growth Suppose that you are studying reproduc
tion in moss. The scaling relation N or L2'll has been found (Niklas, 1994) between the number of moss spores
(N) and the capsule length (L). This relation is not very accurate,
but it turns out that it sufﬁces for your purpose. To estimate the
number of moss spores, you measure the capsule length. If you
wish to estimate the number of moss spores within an error of
5%, how accurately must you measure the capsule length? 48. Tilman’s Resource Model Suppose that the rate of growth
of a plant in a certain habitat depends on a single resource—for
instance, nitrogen. Assume that the growth rate f ( R) depends on
the resource level R in accordance with the formula R
klR where a and k are constants. Express the percentage error of the
growth rate, 1009—}, as a function of the percentage error of the resource level, 1.00%. 49. Chemical Reaction The reaction rate R(x) of the irreversible
reaction f(R)=a A+B—>AB is a function of the concentration x of the product AB and is given
by R(x) = k(a — x)(b — x) where k is a constant, a is the concentration ofA at the beginning
of the reaction. and b is the concentration of B at the beginning
of the reaction. Express the percentage error of the reaction rate, 1009;. as a function of the percentage error of the concentration
x, 100%. 200 Chapter4 I Differentiation Chaptem KeyT‘erms. Discuss the following deﬁnitions and 9. Basic rules of differentiation 18. Derivatives of exponential functions con cepts; 10' Predua rule 19. Derivatives of inverse and 1. Derivative, formal deﬁnition 11. Quotient rule logarithmic functions 2. Difference quotient 12. Chain rule _ . . o 3. Secant line and tangent line 13. Implicit function 20. Logarithmic differentiation
4. Instantaneous rate of change 14. Implicit differentiation 21, Tangent line approximation
5. Average rate of change 15. Related rates . 6. Differential equation 16, Higher derivatives 22 Error Propagallon 7. Differentiability and continuity l7. Derivatives of trigonometric 23. Absolute error, relative error,
8. Power rule functions percentage error Chapter 4 Review Problems ; v . In Problems 1—8, differentiate with respect to the independent (b) Finda line through the origin that touches the graph of f (x) variable. at some point (e. f (c)) with c > 0. This is the tangent line at
4 2 1 (c. f (c)) that goes through the origin. Graph the tangent line in
1 f (x) = ‘3x + 7; + 1 2 g(x) = m the same coordinate system that you used in (a).
1/3 In Problems 26—29, ﬁnd an equation for the tangent line to the curve
(x2 “E he" at the specified point.  3 1 26. y = (sinx)‘°” atx = — 27. y = e“ cosxatx = —
5. f(x) = ez" sin (Ex) 6. g(s) = w 2 3
l f C0503) 28. x2+y=ey atx=./e—1
7_ f(x)=zﬂ%$_). s_ g(x)=ex1n(x+1) 29. xlny=ylnxatx=1 30. In Review Problem 17 of Chapter 2, we introduced the
In Problems 9~1 2, ﬁnd the first and second derivatives of the given following hyperbolic functions: functions 6, _ e_, 9. f(x) = e"‘2/2 10. g(x) = tan(x2 + 1) mm" = 2 v x e R
x 3—: x —x 11' 12‘ f(x)=e—x+1 coshx=e :e , XER In Problems 13—16, ﬁnd dy/dx. ex _ e—x 13. xzy  yzx = sinx 14. ex“yz = 2x mm” = 2! + e:' x E R 15. ln(x  y) = 2x 16. tan(x —— y) = x2 (a) Show that In Problems 1749, ﬁnd dy/dx and dzy/dxz. i sinhx = coshx 17. x2 + y? = 16 18. x = tan y 19. e" = lnx d dx 20. Assume that x is a function oft. Find iiiE when y = cosx and an d if. =ﬁforx= Ecoshx=sinhx I 21. Velocity A ﬂock of birds passes directly overhead. ﬂying (1;) use the factsthat horizontally at an altitude of 100 feet and a speed of 6 feet per Sinhx second. How quickly is the distance between you and the birds mm x = increasing when the distance is 320 feet? (You are on the ground COth and are not moving.) and 22. Find the derivative of cosh2 x — sinh2 x = 1 y = In I cosx I together with your results in (a) to show that d 1
23. Suppose that f (x) is differentiable. Find an expression forthe a tanhx = coshz x
derivative of each of the following functions:
(a) y = em) (b) y = In f(x) (0 y = “(qu 31. Find a seconddegree polynomial
24. Find the tangent line and the normal line to y = lan + l) at p(x) = axz + [7); + c
X = 1.
with p(——l) = 6, p’(1) = 8, and p”(0) = 4.
25. S . th I . .
Uppoge a 32. Use the geometric interpretation of the derivative to ﬁnd thl
x2 equations of the tangent lines to the curve
f(x)= 2.x20 2 2
1 +x x + y = 1 (a) Use a graphing calculator to graph f(x) for x >_ 0. Note that at the following points:
the graph is s shaped. (a) (1. 0) (b) (g. 5/5) __~MM (c) kiﬂiﬁ) (d) (0,—1)
33, Distance and Velocity Geradedort‘3 and Straightville are
connected by a very straight, but rather hilly, road. Biking from
Geradedorf to Straightville, your position at time I (measured in
hours) is given by the function s(t) = 3m + 30— cos(71t)) foro 5 r 5 5.5, where s(r) is measured in miles.
(:1) Use a graphing calculator to convince yourself that you
didn't backtrack during your trip. How can you check this? Assuming that your trip takes 5.5 hobrs, find the distance between
Geradedorf and Straightville. (b) Find the velocity v(t) and the acceleration a(t). (c) Use a graphing calculator to graph 50‘), v(r), and a(t). In (a),
you used the function s(t) to conclude that you didn't backtrack
during your trip. Can you use any of the other two functions to downhill. how many peaks and valleys does this road have? 34. Distance and Velocity Suppose your position at time t on a
straight road is given by s(t) = cos(7rt) for 0 5 t 5 2. where r is measured in hours.
(a) What is your position at the beginning and end of your trip? (b) Use a graphing calculator to help describe your trip in words.
(c) What is the total distance you have traveled? (d) Determine your velocity and your acceleration during the
trip. When is your velocity equal to 0? Relate this velocity to your
position, and explain what it means. 35. Population Growth In one very simple population model, the
growth rate at time 1 depends on the number of individuals at
time I — T, where T is a positive constant. (That is, the model
incorporates a time delay into the birthrate.) This assumption rs
useful, for instance, if one wishes to take into account the fact that
individuals must mature before reproducing. ‘ R I .
(3) Those who are curious may look up the words gerarle and Dorfm a
German—English dictionary. Chapter4 : Review Problems 201 Denote the size of the population at time t by N(r), and
assume that dN 7r
7]? = 57(K — N(t — T)) (4.13) where K and T are positive constants. (a) Show that m
= K A —
N(r) + cos 27, is a solution of (4.13).
(b) Graph N(t) for K = 100. A = 50, and T =1. (c) Explain in words how the size of the population changes over
time. 36. Radoactlve Decay We denote by W(t) the amount of a
radioactive material left at time! if the initial amount present was
W(0) = W0. '
(a) Show that Mr) = woeM solves the differential equation dW (b) Show that ifyou graph W(t) on semilog paper, then the result
is a straight line. (c) Use your result in (b) to explain why d in W(r)
d: Determine the constant, and relate it to the graph in (b).
(d) Show that = constant dln W(t)
=constant
dr
implies that
dW
— Wt
dt o< () 37. Allometric Growth In Example 17 of Subsection 4.4.3, we
introduced an allometric relationship between skull length (in cm) and backbone length (in cm) of ichthyosaurs, a group of extinct
marine reptiles. The relationship is s = (1.162)B°'°” where S and B denote skull length and backbone length,
respectively. Suppose that you found only the skull of an
individual and that, on the basis of the skull length, you wish to
estimate the backbone length of this specimen. How accurately must you measure skull length if you want to estimate backbone
length to within an error of 10%? answer the question of backtracking? Explain your answer. ._ 2 _AW t
a you s ow own gomg uphill and speed up going an. ...
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