Math 1300
1.2 Exponential Functions
Name:
1. Write e5x in the form ax where a is a positive real number.
e5x = e5x = (e5 )x , so a = e5
2. Write 6x in the form ekx where k is a nonzero real number.
6x
Math 1300
Section 2.4
Name: Solutions
1. Suppose that G(v ) equals the fuel eciency, in miles per gallon, of a car going v miles per
hour.
(a) If G(55) = 32.5, What are the units of 55? What are the u
Math 1300
Section 2.5
Name:
Below is a graph of the function f (x) = x3 .
(a) Draw tangent lines on the graph where x = 0.8, 0.5, 0.5, and 0.8.
(b) Use the power rule to write formulas for f (x) and f
Math 1300
2.6 Dierentiability
Name:
1. Decide if these functions are dierentiable at x = 0. Try zooming in on a graphing calculator,
or calculating the derivative f (0) from the denition.
(a) f (x) =
Math 1300
Derivative Practice: 3.1 Powers and Polynomials
In each problem, nd the derivative function.
1. f (x) = x30
f (x) = 30x31
2. f (x) = x4.2
f (x) = 4.2x3.2
3. f (x) = x5/6
f (x) = (5/6)x11/6
1
Math 1300
Derivative Practice through 3.2
In each problem, nd the derivative function.
1. f (x) = 2ex + x2
f (x) = 2ex + 2x
2. f (x) = 5x2 + 4ex
f (x) = 10x + 4ex
3. f (x) = 5x + 2
f (x) = (ln 5)5x
4.
Math 1300
Derivative Problems through 3.2
Choose the correct answer.
1. The function f (x) = 8x is a
function.
(a) linear
(b) power
(c) exponential
2. The function f (x) = 8x has derivative f (x) =
(
Math 1300
3.3 Product & Quotient Rules
Name: Solutions
In each problem, nd the equation of the derivative function using the product or quotient rule.
1. h(x) = (3x )(x4.5 + 5x)
h (x) = (ln 3)(3x )(x4
Math 1300
3.4 The Chain Rule
Name: Solutions
Fill in the blanks.
1. Given
y = (x6 + 4x)20 ,
if y = f (g (x),
then f (z ) =
and z = g (x) =
Now f (z ) =
and g (x) =
Therefore
.
dy
=
dz
dz
=
dx
.
dy
=
d
Math 1300
3.4 The Chain Rule
Name: Solutions
Fill in the blanks.
1. Given
y = (4x + 1)10 ,
if y = f (g (x),
then f (z ) =
and z = g (x) =
Now f (z ) =
and g (x) =
Therefore
.
dy
=
dz
dz
=
dx
.
dy
=
dx
Math 1300
3.5 Word Problem
Name:
A pendulum hung from the ceiling makes a complete back-and-forth swing each 6 sec. As the
pendulum swings, its distance, d cm, from one wall of the room depends on the
Math 1300
Trigonometric Functions and Inverses
Name:
1. Evaluate these expressions exactly. (For example, 3 is exact but 3.1415 is not.)
(a) tan
4
(g) sin (3 )
(b) cos ( )
(h) tan1 (1)
(c) sin
3
3
2
1
Homework 4 (Sections 2.1 to 2.3)
Due in recitation on Tuesday, Sep 20.
1. The displacement (in meters) of a particle moving in a straight line is given by s = t2
where t is measured in seconds.
5t + 1
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Math 1300
Section 2.4
Name:
1. Suppose that G(v ) equals the fuel eciency, in miles per gallon, of a car going v miles per
hour.
(a) If G(55) = 32.5, What are the units of 55? What are the units of 32
Math 1300 2.3 The Derivative Function Name: 301m; uni-
1. Below is a table of values of f Use it to ll in the approximate vaiues of
Hum 6123.3 2 M
Ma) 3 _21
Pm *x 71m? 31 Z
L
# HB
HQ}: 6,8 9
Math 1300
1.3 New Functions from Old
Name:
1. The graph of the function f (x) is shifted left by 3 and up by 10, then stretched vertically by
a factor of 2. Write a formula (involving f ) for the new
Math 1300
Sections 1.5 and 1.6
Name:
1. Evaluate these expressions exactly.
(a) sin
4
(b) tan ( )
(f) cos 56
(g) sin1 (1)
3
2
(c) cos
3
(h) cos1
(d) cos
6
(i) sin1
(e) tan
3
4
(j) tan1 (1)
2
2
2. A si
Math 1300
1.7 Introduction to Continuity
Name:
Sketch graphs of these functions and decide whether or not they are continuous.
You may wish to use your calculator to help you graph.
For each function
Math 1300
Sections 1.8-2.1
Name:
1. Use the gure below, which gives a graph of the function f (x), to give values for the indicated
limits, or indicate if they do not exist.
lim f (x) =
x0
lim f (x) =
Math 1300 Sections 1.8-2.1 Name: Solutions
1. Use the gure below, which gives a graph of the function x}, to give values for the indicated
limits, or indicate if they do not exist.
Ff
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