ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
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Activity C1 – Basic Differentiation Rules with Applications
As we saw in Activity B, the derivative (a.k.a instantaneous rate of change, slope of a
tangent line, or velocity) of a function f ( x) can be found by computing
f '( x)  lim
h 0
f ( x  h)  f ( x)
.
h
Computing this limit, however, can be long and difficult. Thus, we use the derivative
rules(RS1) to make finding derivatives shorter and less complicated.
Use the derivative rules at right to find the following.
1. f '( x) if f ( x)  x2  7 x  5
2.
d 1

 x

dx  x

3. f '(1) if f ( x)  3 x 2 
4.
2
 2x
x3
2
dy
 2.3x 4.1  2 x  6
if y 
3
4
dx
3 x
5. The slope of the line tangent to g ( x)  x2  2x  4 at x  4 .
6. The x value where the slope of the line tangent to g ( x)  x2  2x  4 is zero.
7. The equation of the line tangent to y  3x3  2x2  x  4 at x  1 .
8. Suppose the function s( x)  25 x3  77 x 2  119 x gives the miles a car has traveled in x
hours. If the car started the trip at 4:00pm, find the velocity of the car at t = 1.
9. Which of the following best interprets your answer from problem 8 above?
a) At 5:00pm, the car was traveling backward at a speed of 40 miles per hour.
b) At 5:00pm, the car was traveling forward at a speed of 40 miles per hour.
c) At 1:00pm the car was traveling at a speed of 67 miles per hour.
d) At 5:00pm the car was traveling at an average rate of 67 miles per hour.
e) At 1:00pm the car was traveling at an average rate of 67 miles per hour.
The derivative is a useful application in business, physics, biology, and numerous other
sciences as demonstrated in the next exercise.
ACOW RATES OF CHANGE AND DERIVATIVES MODULE
Updated 5/28/2016
Page 2 of 2
10. The population of a fire ant colony is growing according to the function
N  t   100 t 3  1000 , where t is measured in days. What is the rate of change of the
population with respect to time when t = 9?
11. Which of the following correctly interprets the answer in exercise 10 above?
a) In 9 days, the ant colony had a population of 300 ants.
b) It took 9 days for 300 ants to populate the colony.
c) On the 9th day, the ant colony was increasing at a rate of 300 ants per day.
d) On the 9th day, the ant colony was decreasing at a rate of 300 ants per day.
e) Between the 1st and 9th day, the any colony was increasing at an average rate of
300 ants per day.