Deck 12: Differentiation

Question 1

If the revenue function is given by R(x)= 10x,find the derivative of the revenue function (this is called the marginal revenue).

Answer 1

(x)= 10

Question 2

If the revenue function is given by R(x)= 6x + 4,find the derivative of the revenue function (this is called the marginal revenue).

Answer 2

(x)= 6

Question 3

By the definition of a derivative,the derivative of f(x)= is

A)
B)
C)
D)
E)

Answer 3

Question 4

If the velocity function is given by V(x)= 12x -
,find an equation of the tangent line to the graph of V(x)at

Question 5

By using the definition of a derivative,find f(x)=
.

Question 6

By using the definition of a derivative,find f
(x)where f(x)=
.

Question 7

Find the slope of the tangent line to the curve y =
at the point (0,
).

Question 8

By direct use of the definition of a derivative,find

if f(x)=
.

Question 9

If the profit for a stereo company (in thousands of dollars)is given by the equation
find the slope of the tangent to this curve at (2,33).

Question 10

If the profit for a car company (in thousands of dollars)is given by the equation 400 -
,find the slope of the tangent to this curve at (1,394).

Question 11

If the profit for a toy company (in millions of dollars)is given by the equation P(x)= 4x -
,find the slope of the tangent to this curve at (1,3).

Question 12

If the revenue function is given by R(x)= 11x + 9,find the derivative of the revenue function (this is called the marginal revenue).

Question 13

By direct use of the definition of a derivative,the derivative of f(x)= is

A)
B)
C)
D)
E)

Question 14

By direct use of the definition of a derivative,find

if f(x)=
+ 4.

Question 15

If the profit for a cat food company (in hundred of dollars)is given by the equation P(x)= 178x -
,use the derivative function on your graphing calculator to find the slope of the tangent to this curve at

Question 16

If the profit for a shoe company (in millions of dollars)is given by the equation
find the slope of the tangent to this curve at (1,4).

Question 17

By direct use of the definition of a derivative,find

if f(x)=
- 3x.

Question 18

By direct use of the definition of a derivative,find

if f(x)= 4x - 3.

Question 19

If the revenue function is given by R(x)= 5x - 7,find the derivative of the revenue function (this is called the marginal revenue).

Question 20

If the profit for a telephone company (in millions of dollars)is given by the equation
use the derivative function on your graphing calculator to find the slope of the tangent to this curve at x = 3.69.

Question 21

The demand D for a product at price x is given by D =
+40.Find

and discuss what happens at x = 1.

Question 22

If the velocity function is given by V(x)=
- 22.63,find an equation of the tangent line to the graph of V(x)at (3.51,63.6107).Use your graphing calculator to verity that the graph of V(x)and the tangent line are at the point (3.51,63.6107).

Question 23

When a ball is dropped from 256 feet above the ground,its height H in feet above the ground after t seconds is given by H = 256 - 16
.Find
.

Question 24

When a ball is thrown downward at a speed of 35 feet/s from a height of 1000 feet,its height H in feet after t seconds is given by H = 1000 - 35t -
.Find
.

Question 25

If the velocity function is given by V(x)= 6
+ 13,find an equation of the tangent line to the graph of V(x)at (3,67).

Question 26

The demand D for a product at price x is given by D =
- 1.Find

and discuss what happens at x = 0.

Question 27

When a ball is thrown upward at a speed of 60 feet/s from a height of 500 feet,its height H in feet after
is given by H = 500 + 60t -
.Find
.

Question 28

The demand D for a product at price x is given by D = -5.2
+ 22.Find

and graph the result on your graphing calculator.What is the behavior of the graph when x = 0?

Question 29

If the output function is given by Q(x)=
+ 1320,find the slope of the curve at the point where x = 1.

Question 30

The demand D for a product at price x is given by D =
+ 20.Find

and graph the result on your graphing calculator.What is the behavior of the graph when x = 0?

Question 31

If the output function is given by Q(x)= 5x + 122,find the slope of the curve at the point where x = 7.

Question 32

The derivative of the revenue function (the marginal revenue function)for a product is given by
.Use this information to determine if the revenue equation is continuous for all values of x.

Question 33

If the velocity function is given by V(x)= 7
+ 5x + 1,find an equation of the tangent line to the graph of V(x)at (0,1).

Question 34

If the output function is given by Q(x)= 150 +
,find the slope of the curve at the point where x = 25.

Question 35

If the output function is given by Q(x)=
+ 1,find the slope of the curve at the point where x = 4.

Question 36

If the velocity function is given by V(x)=
+ 12.3,find an equation of the tangent line to the graph of V(x)at (2.47,67.2801).Use your graphing calculator to verity that the graph of V(x)and the tangent line are at the point (2.47,67.2801).

Question 37

The demand D for a product at price x is given by D = 8
.Find

and discuss what happens at

Question 38

The demand for a product is given by p =
.Does the derivative of this function exist for all values of q?

Question 39

The demand D for a product at price x is given by D = -3
+ 100.Find

and discuss what happens at x = 0.

Question 40

If the velocity function is given by V(x)= 3x -
,find an equation of the tangent line to the graph of V(x)at

Question 41

Find the slope of the curve y =
at the point (3,-4).

Question 42

By direct use of the definition of derivative,find
(f(x))if f(x)= 4 - 5
.

Question 43

Find the slope of the curve y =
at the point (3,2).

Question 44

Find the slope of the curve y = 2
at the point (4,4).

Question 45

Suppose the number of calories of heat required to raise 1 gram of water (or ice)from -40°C to x°C is given by
Is this function differentiable for all values of x?

Question 46

Find
if y =
-
+ 7x - 8.

Question 47

Find the slope of the curve y = 4
- 2x + 1 at the point (3,31).

Question 48

If a function is continuous at x = a,then it is differentiable at x = a.

Question 49

If f(x)is defined by f(x)=
,then use your graphing calculator to graph the functions g(x)=
and h(x)= x to determine:
(a)if f(x)is continuous for all values of x.
(b)if f(x)is differentiable for all values of x.

Question 50

Suppose the cost of a product can be approximated by c =
- 2x + 22.Find the derivative of this function to determine whether or not this function is continuous for all values of x.

Question 51

Find
if y =
(9x + 3).

Question 52

Find the slope of the curve y = 3
- x at the point (2,10).

Question 53

If f(x)=
,then find f' (x)and determine whether or not f'(x)exists for all values of x.Use your graphing calculator to graph f(x).Use this graph to describe the behavior of the tangent line to f(x)at

Question 54

By direct use of the definition of derivative,find
(f(x))if f(x)=
.

Question 55

If a function is differentiable at x = a,then it is continuous at x = a.

Question 56

By direct use of the definition of derivative,find
(f(x))if f(x)=
.

Question 57

Find the slope of the curve y =
at the point (11,1).

Question 58

By direct use of the definition of derivative,find
(f(x))if f(x)= 3
+ 5x - 4.

Question 59

Find
if y =
-
+ 4x - 6.

Question 60

By direct use of the definition of derivative,find
(f(x))if f(x)=
.

Question 61

Find
if y =
.

Question 62

Find
if y =
.

Question 63

Find
if y =

-

+
.

Question 64

Find
if y =
-
+
.

Question 65

Find
if y =
+
.

Question 66

Find
if y =
-
.

Question 67

Find
if y =
.

Question 68

Let f(x)=
- 6x + 7.
(a)Find f '(x).
(b)Evaluate f '(2).
(c)Find an equation of the tangent line to the graph of y = f(x)at the point (2,3).

Question 69

Find
if y =
.

Question 70

Find all values of x for which the curve y =
+ 6x - 4 has a horizontal tangent line.

Question 71

Find all values of x for which the curve y = 6
+ 4x - 5 has slope 2.

Question 72

Find
if y =
-
.

Question 73

Let f(x)=
- 3x + 1.
(a)Find
(x).
(b)Evaluate
(1).
(c)Find an equation of the tangent line to the graph of y = f(x)at the point (1,2).

Question 74

Find
if y =
+
.

Question 75

Find
if y =
.

Question 76

Find all values of x for which the curve y = 2
-
has a horizontal tangent line.

Question 77

Find
if y = (0.00
.

Question 78

Find
if y =
+ ln 2.

Question 79

Find
if y =
(x + 2).

Question 80

Find
if y =