Derivatives Text Exercises §3.1: Derivative of a Function

Mathematics 31A
Chapter 3: Derivatives
Text Exercises
§3.1: Derivative of a Function
Worksheet: 1, 4, 6
13, 14, 15, 16, 17b, 22, 26a, 29ab
§3.2: Differentiability
2, 6, 10, 35, 39
Worksheet
§3.3: Rules for Differentiation
12, 15, 21, 24cd, 38
Worksheet
§3.4: Velocity and Other Rates of Change
8, 9ab, 11acd, 12abc, 18, 19 omit f, 24, 32, 37bcdf, 46
§3.5: Derivatives of Trigonometric Functions
3, 4, 8, 30, 31, 36, 37, 38
§3.6: The Chain Rule
13, 17, 19, 21, 32, 53, 58bceg, 59, 63
§3.7: Implicit Differentiation
7, 19, 30, 47, 57
§3.8: Derivatives of Inverse Trigonometric Functions
1, 5, 7, 26, 31
§3.9: Derivatives of Exponential and Logarithmic Functions
5, 7, 11, 13, 17, 19, 21, 41
Review [as needed]
3, 4, 14, 17, 39, 47, 53, 59, 61, 63, 66, 67, 72, 81, 82, 83
Mathematics 31A – Derivatives – Derivative of a Function
f ( x + h) − f ( x)
.
h→0
h
Use the definition of the derivative to find f ′( x) for each of the following.
The derivative of f ( x) is f ′( x) and is defined by f ′( x) = lim
1.
2.
2
x
f ( x) = x − 5
f ( x) = 3 −
It is sometimes useful to find the value of the derivative at a particular value of x. Understand
(algebraically and graphically) that the value of the derivative at x = a can be found using:
f ( a + h) − f ( a )
f ′(a ) = lim
h →0
h
Use the above expression to determine the slope of the tangent line to the given function at
the indicated point.
3. f ( x) = 3 x 2 at x = 12
4.
f ( x) = 2 x − x 2 at x = −1
Understand (graphically) that the value of the derivative at x = a can also be found using:
f ( x ) − f (a )
lim
x →a
x−a
Use the above expression to determine the slope of the tangent line to the given function at
the indicated point.
5. f ( x) = x 2 + 4 at x = −2
6.
Answers
1. x22
2.
1
2 x
3. 3
4. 4
5. –4
6. 14
f ( x) = 1 + x at x = 3
Mathematics 31A – Derivatives – Differentiability
Use your calculator and the nDeriv feature to address each of the following.
1. Given f ( x) = 2 x 2 − x − 1 , determine the value of f ′(−2) .
2. Determine the slope of the line tangent to
( ln x )
f ( x) =
3. Determine the slope of the line normal to f ( x) =
Answers
1. –9
2. 0.139
3. 0.575
2x
3
at x = 2 .
sin ( e x −1 )
x
at x = π .
Mathematics 31A – Derivatives – Rules for Differentiation
Differentiate.
6
x
2x −1
2. g ( x) = 2
x +2
1. y = x 3 −
Given f and g are differentiable functions, write an expression for g ′( x) .
3. g ( x) = x 4 f ( x)
x
4. g ( x) = 2
x + f ( x)
Answers
1. 32 x +
2.
6
x2
−2( x − 2 )( x +1)
( x + 2)
2
2
3. 4 x3 f ( x) + x 4 f ′( x)
4.
− x 2 + f ( x ) − xf ′ ( x )
(x
2
+ f ( x)
)
2
Mathematics 31A – Derivatives – Differentiation Rules Summary
The slope of f at x = a is
f ( a + h) − f ( a )
lim
h →0
h
The derivative of f is
f ( x + h) − f ( x )
lim
h →0
h
d n
x ) = n ⋅ x n −1
(
dx
d
( f ⋅ g ) = f g′ + fg ′
dx
d  f  f g′ − fg ′
 =
dx  g 
g2
d
f ( g ( x) ) = f ′ ( g ( x) ) ⋅ g ′( x)
dx
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec2 x
dx
d
sec x = sec x tan x
dx
d
csc x = − csc x cot x
dx
d
cot x = − csc 2 x
dx
1
d
sin −1 x =
dx
1 − x2
−1
d
cos −1 x =
dx
1 − x2
d
1
tan −1 x =
dx
1 + x2
d x
e = ex
dx
d x
a = a x ln a
dx
d
1
ln x =
dx
x
d
1
log a x =
dx
x ln a
Mathematics 31A – Derivatives – Exam Preparation [optional]
Calculator Active
Traffic flow is defined as the rate at which cars pass through an intersection, measured in
cars per minute. The traffic flow at a particular intersection is modeled by the function F
defined by
t
F (t ) = 82 + 4sin   for 0 ≤ t ≤ 30
 2
where F(t) is measured in cars per minute and t is measured in minutes.
(b) Is the traffic flow increasing or decreasing at t = 7 ? Give a reason for your answer.
(d) What is the average rate of change of the traffic flow over the time interval 10 ≤ t ≤ 15 ?
Indicate units of measure.
Calculator Active
A particle moves along the y-axis so that its velocity v at time t ≥ 0 is given by
v(t ) = 1 − tan −1 ( et ) . At time t ≥ 0 , the particle is at y = −1 .
(a) Find the acceleration of the particle at time t = 2
(b) Is the speed of the particle increasing or decreasing at time t = 2 ? Justify.
Closed Calculator
Consider the closed curve in the xy-plane given by:
x2 + 2 x + y 4 + 4 y = 5
(a) Show that
dy − ( x + 1)
=
.
dx 2 ( y 3 + 1)
(b) Write an equation for the line tangent to the curve at ( −2, 1)
(c) Find the coordinates of any point(s) on the curve where the line tangent to the curve is
horizontal.
(d) Is it possible for the curve to have a horizontal tangent at points where it intersects the xaxis? Explain your reasoning.
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