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 betwe
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
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 .
(
Math 1261: Calculus I
Brown
Implicit Dierentiation and Logarithmic Functions
1. Find
dy
if 2x3 + x2 y = 2 + xy 3 .
dx
2. Dierentiate.
(a) f (x) = ln 8x
(b) g(x) = ln
(x 3)2
ex x2 + sin x
(c) h(x) = [l
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. Probl
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. Dierentia
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 o
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 ta
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
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 v
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
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 o
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 2
Math 1261: Calculus I
Brown
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 vol
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)
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 ca
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
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
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
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) F
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)
li