Section 1.5
1. By imagining tangents at he indicated points state whether the slope is positive,
zero or negative at each point.
P1
P2
P3
2. Use the red tangent lines shown to find
the slopes of the curve at the points of
tangency.
.
P1
.
P2
The tangent line at P 1 contains the points (1, 4) and (2, 6). The
slope between these two points is 2.
The tangent line at P 2 contains the points (5, 4) and (2, 6). The
slope between these two points is -2/3.
3. Find the average rate of change of f (x) = x 2 + x between the following
pairs of x-values.
a.
b.
c.
d.
e.
f.
x=1 and x=3
x=1 and x=2
x=1 and x=1.5
x=1 and x=1.1
x=1 and x= 1.01
What number do your answers seem to be approaching
The average rate of
change is the change
in the y values divided
by the change in the x
values.
OR
You may do this problem using the 4-step procedure.
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 = 2
b. x=1 and x=2 Note h = 1
c. x=1 and x=1.5 Note h = 0.5
d. x=1 and x=1.1 Note h = 0.1
e. x=1 and x= 1.01 Note h = 0.01
f. What number do your answers seem to be approaching
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
f ( x  h)  f ( x) 2xh  h 2  h
Step 4.

 2x  h  1
h
h
Note the beginning x value is 1 for all parts and only the h changes. Plug in x = 1
and the h values at the top of the page into step 4.
2x + h + 1 = 2 + h + 1 = 3 + h
3.1 and e yields 3.01
part a yields 5, b yields 4, c yields 3.5, d yields
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
b. x=3 and x=4
c. x=3 and x=3.5
d. x=3 and x=3.1
e. x=3 and x=3.01
f. What number do your answers seem to be approaching?
The average rate of
change is the change
in the y values divided
by the change in the x
values.
OR
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 = 2
b. x=3 and x=4 Note h = 1
c. x=3 and x=3.5 Note h = 0.5
d. x=3 and x=3.1 Note h = 0.1
e. x=3 and x=3.01 Note h = 0.01
f. 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
Step 4.
f (x  h)  f (x) 5h

5
h
h
Note that by using this method you do not need to do the
problem several time (parts a thru e) 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.
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
f ( x  h)  f ( x) 2xh  h 2  h
Step 4.

 2x  h  1
h
h
Step 5. lim
h0
f (x  h)  f (x)
 lim 2x  h  1  2x  1
h0
h
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 + 5
Step 2:
f (x) = 2x 2 - 3x + 5
Step 3: f (x + h) – f (x) = 4xh + 2h 2 – 3h
f (x  h)  f (x) 4xh  2h 2  3xh
Step 4.

 4x  2h  3
h
h
Step 5. lim
h0
f (x  h)  f (x)
 lim 4x  2h  3  4x  3
h0
h
8. Use the definition of derivative (5-step procedure) to find f ‘ (x) of
f (x) = 9x - 2
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
Step 4.
f (x  h)  f (x) 9h

9
h
h
f (x  h)  f (x)
Step 5. lim
 lim 9  9
h0
h0
h
9. Use the definition of derivative (5-step procedure) to find f ‘ (x) of
f (x) = 4
Step 1: f (x + h) = 4
Step 2:
f (x) = 4
Step 3: f (x + h) – f (x) = 0
Step 4.
f (x  h)  f (x) 0
 0
h
h
f (x  h)  f (x)
Step 5. lim
 lim 0  0
h0
h0
h
10. Use the definition of derivative (5-step procedure) to find f ‘ (x) of
f (x) = 2/x
Step 1: f (x + h) = 2/(x + h)
Step 2:
f (x) = 2/x
Step 3: f (x + h) – f (x) = 0
Step 4 :
f (x  h) – f (x) 
2
2
- 2h
- 
x  h x x (x  h)
- 2h
f (x  h)  f (x)
-2
x (x  h)
Step 5. lim
 lim
 lim
h0
h0
h  0 x (x  h)
h
h

2
  2x  2
2
x
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
Step 4.
xh 
h
f (x  h)  f (x)

h

h
h( xh 
x)
x

f (x  h)  f (x)
Step 5. lim
 lim
h0
h0
h

xh 
h
x
xh 
x
xh 
x
1
xh 
1
xh 
x
1 12

 x
x 2 x 2
1
Hint Multiply the numerator or denominator of the difference quotient by
√(x + h) + √x and then simplify.
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.
Use a graphing calculator to graph the curve together with the tangent line to
verify your answer.
See problem 6.
13. In problem 8 you found the derivative of f (x) = 9x – 2. Explain why you answer
makes sense.
The derivative was 9. That makes sense because the derivative is the slope of the
curve and the curve in problem 8 is a straight line with slope 9.
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 interpret 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 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 person
could 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 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 States
who 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 = 6 would stand for 1990).
a. Find f ’(x) using the definition of the derivative.
b. Evaluate the derivative at x=1 and interpret the result
c. Find the rate of change of the immigrant percentage in the year 2000.
a. Use the five-step procedure 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 %.