Limits and Continuity of Functions of Two or More Variables
Introduction Recall that for a function of one variable, the mathematical statement
means that for x close enough to c, the difference between f(x) and L is "small". Very similar definitions exis
Math 1261: Calculus I
Brown
The Area Problem
1. Estimate the area under the graph of f (x) = x from x = 0 to x = 4 using four approximating
rectangles.
(a) Use right endpoints. Is this an overestimate or an underestimate? Why?
(b) Use left endpoints. Is t
Math 1261: Calculus I
Brown
Antiderivatives
1. Find all the antiderivatives of the following.
(a) f (t) = 8t9 3t6 + 12t2
(b) g(x) = 2
(c) h(x) = 4ex sec x tan x
t 8t3 + t e
(d) j(t) =
t2
2. Find f .
(a) f (x) = 3x 3
(b) f (x) = 2x 3/x4 ,
x > 0,
3
(c) f (t
Math 1261: Calculus I
Brown
Limits Precisely
1. (a) Find the number such that if 0 < |x 6| < , then | 3x + 18| < , where = 0.1.
(b) Repeat (a) with = 0.01.
(c) Prove that lim (3x + 8) = 10
x6
2. Problem 2.7.42.
3. Prove that lim
x
1
= 0.
x
|x|
, Using the
Math 1261: Calculus I
Brown
Derivatives of Trigonometric Functions
1. Use the quotient rule to establish the following derivative formulas.
d
tan x = sec2 x
dx
d
(b)
cot x = csc2 x
dx
(a)
2. Dierentiate f (x) =
d
sec x = sec x tan x
dx
d
(d)
csc x = csc x
Math 1261: Calculus I
Brown
Limits and Innity
1. Is x = 3 is a vertical asymptote of g(x) =
x2 + 4x 21
? Why or why not?
x2 x 6
2. Explain what is wrong with the statement, 5 is a vertical asymptote of r(x) =
3. Find the vertical and horizontal asymptote(
Math 1261: Calculus I
Brown
Derivatives
1. (a) Write the denition of the derivative of f at a.
(b) Use this denition to compute the derivative of f (x) =
1
at 2.
x
(c) Write an equation of the line tangent to f at 2.
2. Problem 3.1.53
3. Prove that if f i
Math 1261: Calculus I
Brown
Related Rates
1. If V is the volume of a sphere of radius r, and the sphere expands as time passes, nd dV /dt in terms
of dr/dt.
2. The length of a rectangle is increasing at a rate of 6 cm/s and the width is decreasing at a ra
Math 1261: Calculus I
Brown
Extrema, Derivatives, and Graphs
1. Find the critical numbers of the following functions.
(a) f (x) = x3 + x2 + x
(b) g(x) = 1 x2
2. Find the absolute maximum and minimum values of f (x) = ex
3
x
on [1, 0].
3. The graph of the
Math 1261: Calculus I
Brown
Continuity
1. Problem 2.6.11.
2. (a) Explain why R(x) = ln x +
Theorems 2.10 and 2.14.]
x3
is continuous at every number in its domain. [Hint: Use
2x2 7x + 3
(b) State the domain of R.
(c) Find the discontinuities of R.
(d) Wha
Math 1261: Calculus I
Brown
Graphs and Derivatives
1. The graph of the derivative f of a continuous function f is given below.
1
y = f (x)
(a) Find the critical numbers of f .
(b) Find the intervals over which f is increasing or decreasing.
(c) Find the n
Math 1261: Calculus I
Brown
Limit Yoga
1. Problem 2.2.20.
2. Evaluate the following limits, if they exist.
x2 5 2
x3
x3
2t + 7
t1
(d) lim
+
2 + 5t + 6
t2 t
t+2
x2 4x
2 3x 4
x4 x
u2 2u 3
(b) lim
u1/2 2u2 + 3u + 1
(a) lim
2x 1
4
3. Let f (x) =
2
x 4
4. Le
Math 1261: Calculus I
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Optimization Problems
1. Find a positive number such that the sum of it and its reciprocal is as small as possible.
2. A box with a square base and open top must have a volume of 32,000 cm3 . Find the dimensions of the
box that
Math 1261: Calculus I
Brown
Precalculus Review
Functions
1. Write the denition of function.
2. Suppose f (x) = 2x2
x
+ 1. Evaluate the following.
3
(c) f (x + h)
f (x + h) f (x)
(d)
,
h
(a) f (3)
(b) f (5) f (2)
h=0
3. State, in order, the transformation
Math 1261: Calculus I
Brown
Linearization and Dierentials
1. Suppose f is dierentiable at a.
(a) Write an equation of the line tangent to y = f (x) at a.
(b) Solve for y and call this La (x). La is called the linearization of f at a.
2. Suppose f (x) = x.
Math 1261: Calculus I
Brown
The Fundamental Theorem of Calculus
1. Evaluate.
(a)
1
d
dx
1
earctan t dt
(b)
0
0
d arctan x
(e
) dx
dx
(c)
2. Dierentiate the following.
x
(a) F (x) =
tan t dt
0
4
arcsin t
dt
t2 + 5
x
cos x
1 t2
dt
(c) H(x) =
t2 + 2
1/2
(b)
Math 1261: Calculus I
Brown
Average Value of a Function
1. Find the average value of f (x) = 4 x2 on [0, 2].
2. Mean Value Theorem for Integrals: Let f be continuous on a closed interval [a, b]. Then there
exists c in (a, b) such that
b
1
f (c) =
f (x) dx
Triple Integrals
Integration of a function of three variables, w=f(x,y,z), over a threedimensional region R in xyz-space is called a triple integral and is denoted
Triple Integrals in Box-Like Regions Suppose that R is the box with a<=x<=b, c<=y<=d, and r
V ; reprebonmhovn of wahqns
v vevwoly~ by a desormh'on )h wovd
d, hvme/Vncany by «mac 0? vmues
-; VIQUOIHY bag 0} rapid
~ a\5ebrmical|y«by 01h eXphcfbvmuM
mamaca Find «ne Aommm 0? each
fury/e
3 (a) Fm=JX+Z 3 (S900: \
z Xzx
TSOUOH Povcm) b/c ne square mo
Math 1261: Calculus I
Brown
LHospitals Rule
1. Evaluate the following limits.
ln x
sin(x)
x2 + 2
(b) lim
x
2x2 3
x
e
(c) lim n , n is a positive integer.
x x
sin x
(d) lim
x 1 cos x
(a) lim
x1
(e)
lim x2 ex
x
(f) lim x tan(1/x)
x
(g) lim+ csc x
x0
1
x
(