1, For the given function find the average rate of change over each specified interval.

f(x) =x²+x12

(a)[0,2]

(b)[1, 10]

2,Givenf(x) =x²+2x+6,

find the average rate of change off(x)

over each of the following pairs of intervals.

(a)

[1.9, 2] and [1.99, 2]

average rate of change over [1.9, 2]average rate of change over [1.99, 2]

(b)

[2, 2.1] and [2, 2.01]

average rate of change over [2, 2.1]average rate of change over [2, 2.01]

(c)

What do the calculations in parts (a) and (b) suggest the instantaneous rate of change off(x)

atx= 2

might be?

3,We are givenf(x) =9x²andf'(x) =18x.

(a) Find the instantaneous rate of change off(x)

atx=2.

(b) Find the slope of the tangent to the graph ofy=f(x)

atx=2.

(c) Find the point on the graph ofy=f(x)

atx=2.

(x,y) =

We are givenf(x) =8x²

andf'(x) =16x.

Exercise (a)

Find the instantaneous rate of change off(x)

atx=3.

Exercise (b)

Find the slope of the tangent to the graph ofy=f(x)

atx=3.

Exercise (c)

Find the point on the graph ofy=f(x)

atx=3

5,In Example 6 in this section, we were givenf(x) = 3x²+ 2x+ 11

and foundf'(x) = 6x+ 2.

(a)

Find the instantaneous rate of change off(x)

atx=9.

(b)

Find the slope of the tangent to the graph ofy=f(x)

atx=9.

(c)

Find the point on the graph ofy=f(x)

atx=9.

(x,y) =

6,Letf(x) =4x²2x.

(a) Use the definition of derivative and the Procedure/Example box in this section to find the derivative.

f'(x) =

(b) Find the instantaneous rate of change off(x) atx=1.

(c) Find the slope of the tangent to the graph ofy=f(x)

atx=1.

(d) Find the point on the graph ofy=f(x)

atx=1.

(x,y) =

7,The tangent line to the graph off(x) atx=1

is shown. On the tangent line,Pis the point of tangency andAis another point on the line.Thexy-coordinate plane is given. There is 1 curve and 1 line on the graph.

• The curve enters the window in the second quadrant, goes down and right becoming less steep, passes through the point(2, 0.5)crossing the line, crosses thex-axisat approximatelyx=1.7,becomes nearly horizontal at the approximate point(0,0.8),goes down and right becoming more steep, passes through the point(1,1)touching the line, and exits the window in the fourth quadrant.
• The line enters the window in the second quadrant, goes down and right, passes through the point(2, 0.5)crossing the curve, crosses thex-axisatx=1,crosses they-axisaty=0.5,passes through the point(1,1)touching the curve, and exits the window in the fourth quadrant.
• PointPoccurs at(1,1).
• PointAoccurs at(3,2).

(a) Find the coordinates of the pointsPandA.

P(x,y)=A(x,y)=

(b) Use the coordinates ofPandAto find the slope of the tangent line. (Give an exact answer. Do not round.)

(c) Findf'(1).

(Give an exact answer. Do not round.)

(d) Find the instantaneous rate of change off(x) atP. (Give an exact answer. Do not round.)

8The tangent line to the graph off(x)

atx= 1

is shown. On the tangent line,Pis the point of tangency andAis another point on the line.Thexy-coordinate plane is given. There is 1 curve and 1 line on the graph.

• The curve enters the window in the third quadrant, goes up and right becoming less steep, crosses they-axisaty=2,changes direction at the point(1,1)touching the line, goes down and right becoming more steep, and exits the window in the fourth quadrant.
• The line enters the window in the third quadrant, goes horizontally right, crosses they-axisaty=1,passes through the point(1,1)touching the curve, and exits the window in the fourth quadrant.
• PointPoccurs at(1,1).
• PointAoccurs at(0,1).

(a) Find the coordinates of the pointsPandA.

P(x,y)=A(x,y)=

(b) Use the coordinates ofPandAto find the slope of the tangent line.

(c) Findf'(1).

(d) Find the instantaneous rate of change off(x)

atP.

9Consider the following.

f(x) =3x²+6x19

(a) Find the derivative, by using the definition.

f'(x) =

(b) Find the instantaneous rate of change of the function at any value.

f'(x) =

Find the instantaneous rate of change of the function at the valuex=4.

f'(4) =

(c) Find the slope of the tangent at the valuex=4

10,Consider the following.

p(q) =2q²+q+7

(a) Find the derivative, by using the definition.

p'(q) =

(b) Find the instantaneous rate of change of the function at any value.

p'(q) =

Find the instantaneous rate of change of the function at the valueq=11.

p'(11) =

(c) Find the slope of the tangent at the valueq=11.

11,Use the given table to approximatef'(a) as accurately as you can.

f'(13) =x12.012.991313.1a= 13f(x)1.4717.4217.1722.86

12,In the figure, at each pointAandBdraw an approximate tangent line and then use it to answer the following questions.Thexy-coordinate plane is given. The curvef(x)enters the window in they-axisaty= 4,goes up and right becoming less steep, passes through the approximate pointA(2, 7.7),changes direction at the approximate point(4, 9.1),goes down and right becoming more steep, passes through the approximate pointB(5, 8.8),and exits the window in the first quadrant.

(a) Isf'(x)

greater at pointAor at pointB? Explain.

f'(x) is greater at pointA. The slope of the tangent line is positive atA.

f'(x) is greater at pointA. The slope of the tangent line is negative atA.

f'(x) is greater at pointB. The slope of the tangent line is positive atB.

f'(x) is greater at pointB. The slope of the tangent line is negative atB.

(b) Estimatef'(x)

at pointB.

0

3

1

3

3

1

3