Worksheet for Section 2.4 The Product and Quotient Rules
V63.0121, Calculus I Spring 2010
1. Evaluate the derivative of each of the following functions. (i) y = x sin(x)
(ii) f (t) = sin(t) + sin( ) + te
(iii) x(t) =
t3 2t + 4 cos(t) + 2
(iv) y =
sin(x) c
Solutions to Worksheet for Section 4.5 Optimization Problems
V63.0121, Calculus I Spring 2010
1. An advertisement consists of a rectangular printed region plus 1 in margins on the sides and 2 in margins on the top and bottom. If the area of the printed re
Worksheet for Section 4.5 Optimization Problems
V63.0121, Calculus I Spring 2010
1. An advertisement consists of a rectangular printed region plus 1 in margins on the sides and 2 in margins on the top and bottom. If the area of the printed region is to be
Worksheet for Section 4.4 Curve Sketching
V63.0121, Calculus I Spring 2010
Graph the functions completely, labeling all signicant points. 1. f (x) = x(6 2x)2
2. f (x) = x4 + 6x2 4
3. f (x) = x 8 x2
1
4. f (x) = xe1/x
5. f (x) = x ln(x2 )
6. f (x) = cos x
Worksheet for Section 4.2 Derivatives and the Shapes of Curves
Spring 2010
1. Let f (x) = x3 6x2 9x + 2. (a) Find the intervals on which f is increasing and decreasing (b) Use the rst derivative test to nd the local maxima and minima (c) Find the interval
Worksheet for Section 4.2 The Mean Value Theorem
V63.0121, Calculus I Spring 2010
1. For each of the functions f and intervals [a, b] below, try to nd a c for which f (c) = f (b) f (a) . If c cannot be found, explain why the Mean Value Theorem is not cont
Worksheet for Section 4.1 Maximum and Minimum Values
V63.0121, Calculus I Spring 2010
1. Decide whether each labeled point is an absolute maximum, an absolute minimum, a relative maximum, a relative minimum, or none of these: y B E C F D x G
A 2. Find the
Worksheet for Section 3.4 Exponential Growth and Decay
V63.0121, Calculus I Spring 2010
1. The half-life of Uranium-235 is approximately 7 108 years. If 50 g are buried at a nuclear waste site, how much will remain after 100 years?
2. A population of E. C
Worksheet for Section 2.6 Implicit Dierentiation
V63.0121, Calculus I Spring 2010
For each of the two problems below: (a) Find dy/dx by dierentiating implicitly. (b) Solve the equation for y as a function of x, and nd dy/dx explicitly from that equation (
Worksheet for Section 5.1 Areas and Distances
V63.0121, Calculus I Spring 2010
1. Draw the graph of f (x) = ex on the interval [0, 1]. We are going to nd the area below the curve on this interval.
2. Estimate the area by computing with a calculator. (i) L
Solutions to Worksheet for Section 5.1 Areas and Distances
V63.0121, Calculus I Spring 2010
1. Draw the graph of f (x) = ex on the interval [0, 1]. We are going to nd the area below the curve on this interval. Solution. Here is the region: y 3 2 1 x 1
2.
Worksheet for Section 2.3 Basic Dierentiation Rules
V63.0121, Calculus I Spring 2010
1. Evaluate the derivative of each of the following functions. (i) y = 10 sin(x) + 6
(ii) h(x) = (2x 3)(x + 2)
(iii) y =
1 1 x2 x
(iv) y =
3
t2
4
t3
a (v) y = x2 4 x3
Worksheet for Section 2.12 The Derivative and Rates of Change The Derivative as a Function
V63.0121, Calculus I Spring 2010
1. Let f (x) = x3 . Use the denition of the derivative to nd f (2).
2.
Let f (x) =
x.
(a) Use the denition of the derivative to nd
Worksheet for Section 1.6 Limits involving Innity
V63.0121, Calculus I Spring 2010
1. Graph 2x3 16 , x3 27 after calculating the limits as x 3 and x . What are the vertical and horizontal asymptotes. y=
2. Let t be the time in weeks. At time t = 0, organi
Solutions to Worksheet for Section 5.5 Integration by Substitution
V63.0121, Calculus I Spring 2010
Find the following integrals. In the case of an indenite integral, your answer should be the most general antiderivative. In the case of a denite integral,
Worksheet for Section 5.5 Integration by Substitution
V63.0121, Calculus I Spring 2010
Find the following integrals. In the case of an indenite integral, your answer should be the most general antiderivative. In the case of a denite integral, your answer
Section 5.4 The Fundamental Theorem of Calculus
V63.0121, Calculus I Spring 2010
Compute the derivatives of the following functions.
x
1. g (x) =
0
sin t dt
Solution. By the Fundamental Theorem of Calculus, g (x) = sin x.
3x
2. g (x) =
0
sin t dt
u
Soluti
Section 5.4 The Fundamental Theorem of Calculus
V63.0121, Calculus I Spring 2010
Compute the derivatives of the following functions.
x
1. g (x) =
0
sin t dt
3x
2. g (x) =
0
sin t dt
0
3. g (x) =
2x
sin t dt (Hint: reverse the order of the integral.)
3x
4.
Worksheet for Section 5.3 Evaluating Denite Integrals
V63.0121, Calculus I Spring 2010
Evaluate the following denite integrals.
4
1.
1
x3
x dx
2.
0
(sin x + 2 cos x) dx
1
3. A company is considering a new product and would like to understand the costs of
Worksheet for Section 2.5 The Chain Rule
V63.0121, Calculus I Spring 2010
1. Dierentiate each of the following functions. 1. y = (x3 x + 1)5
2. f (t) =
3
1 + tan t
3. y = a3 + sin3 x
4. y = sin(a3 + x3 )
1
2. If r(x) = f (g (h(x) and h(1) h (1) g (2) g (