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Unformatted text preview: MATH 125 : Review 2 Problem 1 Consider the following function $2 + 311: + 1
Its ﬁrst and second derivatives are given below:
3(1 — $2) 6(233 — 3x)
I _ //
f(.'13) (£132 +1)2, f (3:) ($2 +1)3 1) Give the domain of f. Determine if f is odd, even or none of these. 2) Find the limits of f where needed. Write the equations of the asymp—
totes of f, if any. 3) Determine where f is increasing, decreasing. Find the x and y 
coordinates of the local maxima and minima of f. 4) Determine where f is concave upwards, downwards. Find the x
coordinates of the inﬂection points. 5) Find the y—intercept.DraW a rough sketch of the graph of f. It should
reflect all the data you found. Problem 2 Consider f (:r) = (1 +13”. a) Find the linear approximation of f at a = 2. 1 b) Use a) to ﬁnd an approximate value of W' C) Is that approximation an overestimate or a underestimate of the actual L?
value of 3.14 . Problem 3 f (2:) = 3334 — 8253 + 6x2
1) ﬁnd all the critical points of f and classify them as local max or local min.
2) ﬁnd the xy coordinates of the global maxima and global minima of f on
the interval [—1, 2]. Problem 4
Given the graph of the derivative function f’ below, answer the following
questions: ' a) Indicate the m—coordinates of the critical points of f. b) Indicate on which intervals f is increasing/ decreasing and the LII—coordinates
of the local maxima/minima if there are any. 0) Indicate on which intervals f is concave up / down and the {II—coordinates
of the inﬂection points if there are any. (1) Sketch a potential graph for f knowing f (0) = O. Problem 5 Consider f(t) = tsint  t for 0 S t S g.
a) Compute f’ (t) b) Prove that the graph of f has at least one horizontal tangent line in
the interval [0, g . (Do not try to solve f’ it equal to 0). c) Prove, that the graph of f has ONLY ONE horizontal tangent line in
the interval [0, £5 . Problem 6
A storage tank has the shape of a cylinder with ends capped by 2 ﬂat disks.
The price of the top and bottom caps is 3 dollars/ square meter. The price
of the cylindrical wall is 2 dollars/ square meter. What are the dimensions
of the cheapest storage tank that has volume of 1 cubic meter. Problem 7 Consider f (:r) = x — xln(x) a) Show that f is strictly decreasing in the interval (1,00) b) Show that f has an inverse function (we will denote it by f ‘1) on the
interval (1,00). c) Evaluate (f‘1)’(0) and write the equation of the tangent line to f “1
at a: = 0. Problem 8
a: f(m) = (a: + 3)2 1) Give the domain of f. Determine if f is odd, even or none of these. 2) Find the limits of f where needed. Write the equations of the asymp
totes of f, if any. 3) Determine where f is increasing, decreasing. Find the x and y 
coordinates of the local maxima and minima of f. 4) Determine where f is concave upwards, downwards. Find the x
coordinates of the inflection points. 5) Find the y—intercept. Draw a sketch of the graph of f. Problem 9 Consider f (x) = 633+”
a) Find the equation of the tangent line of f at a = 0.
b) Use a) to ﬁnd an approximate value of f (—0.02). c) Is that approximation an overestimate or a underestimate of the actual
value of f (—0.02) (Justify without using your calculator)? Problem 10 f(:1:) = (1 — 2:1: — 2:2)e_z 1) ﬁnd all the critical points of f and classify them as local max or local
min. 2) ﬁnd the xy coordinates of the global maxima and global minima of f
on the interval [0, 2]. Problem 11
Given the graph of the derivative function f’ below, answer the following
questions: 1 a) Indicate the x—coordinates of the critical points of f. b) Indicate on which intervals f is increasing/ decreasing and the m—coordinates
of the local maxima/minima if there are any. 0) Indicate on which intervals f is concave up / down and the xcoordinates
of the inﬂection points if there are any. (1) Sketch a potential graph for f. Problem 12
Consider f(a:) = mln(x) — a: for :1: in [1, e] a) Show, Without computing the derivative f’, that the equation
f’ (II?) = 8+1 has at least one solution. b) Now solve the equation f’ (2:) = —. e—l Problem 13 Find the derivatives of the following functions: a) f(:I:) = ln(1+ ems) d) f (33) = 0083(ln($)) 6) f(1‘) = (63 + 1V? f) gov) = —;{:::)1; _ gI«‘(2~2)(av+1)3
9 W)  W Problem 14
An oval running track is to be built by adding semicircles onto each end of
a rectangle, as shown. The total perimeter of the track must be 400 meters,
and the length L of each of the straight segments must be between 90 and
120 meters. Find the value of L which will
a) maximize the area of the rectangle;
b) minimize the area of the rectangle. Problem 15 Consider f (x) = x2 + 69:2
a) Show that f is strictly increasing in the interval (0,00) b) Show that f has an inverse function (we will denote it by f ‘1) on the
interval (0,00). c) Evaluate (f‘1)’ (1 + e) and write the equation of the tangent line to
f '1 at x = 1 + 6. Problem 16 1) Find the equation of the tangent line to the curve given by :1: + 6%? +
y3 = 6 at the point (2,0). 2) Given the curve y3 — y = (:E + l)2, ﬁnd the concavity of the curve at
the point (—1,1). Problem 17 A 13 ft long ladder leans against a wall. It starts to slip. When the base
of the ladder is 5 ft from the wall, it is sliding away from the wall at a rate
of 4 ft per second. Find how fast the top of the ladder is sliding down the
top of the wall when the base of the ladder is 5 feet away from the wall. Problem 18 10000 cubic meters of oil is spilled into the ocean. The oil takes the shape of
a cylinder. The radius of the cylinder increases at a rate of 4 meters/ hour.
(Recall the volume of a cylinder is given by V = 7rr2h where 7“ is the radius
and h is the height or thickness). At what rate is the thickness of the oil spill decreasing when the radius is
100 meters? (We assume that the oil and the water are not mixing so the
volume of the cylinder remains constant) Problem 1 9 Find the area of the largest rectangle that can be drawn with the left side along the y—axis, the bottom side along the xaxis and the upper right corner on the curve y = 31%;. (see below) ...
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 Summer '07
 Tuffaha
 Calculus, Derivative

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