Section 2.2

1. By imagining tangents at he indicated points state whether the slope is positive, zero or negative at each point.

P 1P 2

P 3

2. Use the red tangent lines shown to find the slopes of the curve at the points of tangency.

..

3. Find the average rate of change of f (x) = x 2 + x between the following pairs of x-values.

a. x=1 and x=3b. x=1 and x=2c. x=1 and x=1.5d. x=1 and x=1.1e. x=1 and x= 1.01f. What number do your answers seem to be approaching

3. Find the average rate of change of f (x) = x 2 + x between the following pairs of x-values. a. x=1 and x=3 Note h = 2b. x=1 and x=2 Note h = 1c. x=1 and x=1.5 Note h = 0.5d. x=1 and x=1.1 Note h = 0.1e. x=1 and x= 1.01 Note h = 0.01f. What number do your answers seem to be approaching

You may do this problem using the 4-step procedure.

Step 1: f (x + h) = (x + h) 2 + ( (x + h) = x 2 + 2xh + h 2 + x + h

Step 2: f (x) = x 2 + x

Step 3: f (x + h) – f (x) = 2xh + h 2 + h

1hx2h

hhxh2

h

)x(f)hx(f.4Step

2

Note the beginning x vale is 1 for all parts and only the h changes. Plug in those values to step 4.

2x + h + 1 = 2 + h + 1 = 3 + h = ?? part a yields 5, b yields 4, c yields 3.5, d yields 3.1 and e yields 3.01

4. Find the average rate of change of f(x) = 5x +1 between the following pairs of x-values. a. x=3 and x=5b. x=3 and x=4 c. x=3 and x=3.5d. x=3 and x=3.1e. x=3 and x=3.01f. What number do your answers seem to be approaching?

You may do this problem using the 4-step procedure.4. Find the average rate of change of f(x) = 5x +1 between the following pairs of x-values. a. x=3 and x=5 Note h = 2b. x=3 and x=4 Note h = 1c. x=3 and x=3.5 Note h = 0.5d. x=3 and x=3.1 Note h = 0.1e. x=3 and x=3.01 Note h = 0.01f. What number do your answers seem to be approaching?

Step 1: f (x + h) = 5(x + h) + 1 = 5x + 5h + 1

Step 2: f (x) = 5x + 1

Step 3: f (x + h) – f (x) = 5h

f (x h) f (x) 5hStep 4. 5

h h

Note that by using this method you do not need to do the problem several time (parts a thru g) to get the value of 5.

5. Find the instantaneous rate of change of f (x) = x 2 + x at x = 1. Use the five step procedure.

h 0 h 0

f (x h) f (x)Step 5. lim lim 2x h 1 2x 1

h

Step 1: f (x + h) = (x + h) 2 + (x + h) = x 2 + 2xh + h 2 + x + h

Step 2: f (x) = x 2 + x

Step 3: f (x + h) – f (x) = 2xh + h 2 + h

1hx2h

hhxh2

h

)x(f)hx(f.4Step

2

f ’ (1) = 2 (1) + 1 = 3

6. Find the slope of the tangent of 2x 2 + x – 2 at x = 2

Graph this on your calculator and use “draw” “tangent” to get the answer. Remember the slope of the tangent is the number in front of the x in the tangent equation. In this case the slope is 9.

7. Use the definition of derivative (5-step procedure) to find f ‘ (x) of f (x) = 2x 2 – 3x + 5

Step 1: f (x + h) = 2(x + h) 2 - 3(x + h) + 5 =2 x 2 + 4xh + 2h 2 - 3x – 3h + 5Step 2: f (x) = 2x 2 - 3x + 5

Step 3: f (x + h) – f (x) = 4xh + 2h 2 – 3h

2f (x h) f (x) 4xh 2h 3xhStep 4. 4x 2h 3

h h

h 0 h 0

f (x h) f (x)Step 5. lim lim 4x 2h 3 4x 3

h

Step 1: f (x + h) = 9(x + h) – 2 = 9x + 9h - 2

Step 2: f (x) = 9x - 2

Step 3: f (x + h) – f (x) = 9h

f (x h) f (x) 9hStep 4. 9

h h

h 0 h 0

f (x h) f (x)Step 5. lim lim 9 9

h

8. Use the definition of derivative (5-step procedure) to find f ‘ (x) of f (x) = 9x - 2

Step 1: f (x + h) = 4

Step 2: f (x) = 4

Step 3: f (x + h) – f (x) = 0

f (x h) f (x) 0Step 4. 0

h h

h 0 h 0

f (x h) f (x)Step 5. lim lim 0 0

h

9. Use the definition of derivative (5-step procedure) to find f ‘ (x) of f (x) = 4

Step 1: f (x + h) = 2/(x + h)

Step 2: f (x) = 2/x

Step 3: f (x + h) – f (x) = 0

h 0 h 0 h 0

f (x h) f (x)Step 5. lim lim lim

h h

- 2h- 2x (x h)

x (x h)

10. Use the definition of derivative (5-step procedure) to find f ‘ (x) of f (x) = 2/x

h)(xx

2h-

x

2-

hx

2 (x) f– h) (x f:4 Step

22 x2

x

2

11. Use the definition of derivative (5-step procedure) to find f ‘ (x) of f (x) = √x

Step 1: f (x + h) = √(x + h)

Step 2: f (x) = √x

Step 3: f (x + h) – f (x) = √(x + h) - √x

x h x x h x x h xf (x h) f (x)Step 4.

h h h x h x

12

h 0 h 0

f (x h) f (x) 1 1 1Step 5. lim lim x

h 2x h x 2 x

Hint Multiply the numerator or denominator of the difference quotient by √(x + h) + √x and then simplify.

h 1

h ( x h x) x h x

12. Find the equation to the tangent line to the curve f (x) = x 2 – 3x + 5 at x = 2, writing the equation in slope intercept form.

b. Use a graphing calculator to graph the curve together with the tangent line to verify your answer.

See problem 6.

13. a. Find f’(x) using the definition of the derivative. b. Explain, by considering the original function, why the derivative is a constant.

f (x) = 3x - 4

b.

Use the five step procedure. See problem 8.

f ‘ (x) = 3.

14. Business: Temperature The temperature in an industrial pasteurization tank is f (x) = x 2 – 8x + 110 degrees centigrade after x minutes (for 0 ≤ x ≤ 12)

a. Find f’(x) by using the definition of the derivative.b. Use your answer to part (a) to find the instantaneous rate of change of the temp.after 2 minutes. Be sure to interrupt the sign of your answer.c. Use your answer to part (a) to find the instantaneous rate of change after 5 minutes.

a. Use the five-step procedure (See problem 7) to get

f ’ (x) = 2x – 8.

b. f ’ (2) = 2 (2) – 8 = - 4 After two minutes the temperature is decreasing at a rate of 4 degrees per minute.

c. f ’ (5) = 2 (5) – 8 = 2 After five minutes the temperature is increasing at a rate of 2 degrees per minute.

15. Behavioral Science: Learning theory In a psychology experiment, a personcould memorize x words in 2x 2 -x seconds (for 0 ≤ x ≤ 10 )a.Find f’(x) by using the definition of the derivative.b.Find F’(5) and interpret it as an instantaneous rate of change in the proper units.

a. Use the five-step procedure (See problem 7) to get

f ’ (x) = 4x – 1.

b. f ’ (5) = 4 (5) – 1 = 19 it will take 19 seconds to memorize the next, sixth, word..

16. Social Science: Immigration The percentage if people in the United Stateswho are immigrants (that is, born elsewhere) for different decades is approximated by the function f (x) = ½ x 2 – 3.7 x + 12, where x stands for the number of decades since 1930 (so that, for example x=5 would stand for 1990).

a. Find f ’(x) using the definition of the derivative.b. Evaluate the derivative at x=1 and interpret the resultc. Find the rate of change if the immigrant percentage in the year 2000.

a. Use the five-step procedure (See problem 7) to get

f ’ (x) = x – 3.7.

b. f ’ (1) = 1 – 3.7 = - 2.7 During the 1940’s the percentage of immigrants was decreasing by 2.7 %.

c. f ’ (7) = 7 – 3.7 = 3.3 During the 2000’s the percentage of immigrants was increasing by 3.3 %.