AB Calculus AB
Review: Meaning of Derivative
Name _______________________
1. A spaceship approaches a far-off planet. At time x minutes after its retrorockets fire, its distance f(x) in kilometers
2
from the surface of the planet is given by f ( x)  x  8x  18 .
(a) Find the average rate of change of f(x) with respect to x from x = 5 to x = 6. What are the units of this rate of
change?
(b) Estimate the rate of change at x = 5.
2. What does derivative mean? What is its formal definition?
3. Use the definition of derivative to calculate f'(x).
(a)
f ( x)  x2  5x  1
(b) f(x)=5
(c)
f ( x) 
4. Using the graph, answer the following questions.
i) Where is the derivative 0?
ii) Over which interval is the graph discontinuous?
iii) Where does the derivative not exist and why?
iv) Over which interval(s) is the derivative positive?
v) Over which interval(s) is the derivative negative?
vi) Over which interval(s) is the 2nd derivative positive?
vii) Over which interval is the 2nd derivative negative?
1
x2
(d) f ( x ) 
x 1
5. Label the following: h,
f(x+h),
f(x),
f(x+h)-f(x), segments with slopes
6. Given the graphs below, sketch the derivative.
f ( x  h)  f ( x )
h
f ( x  h)  f ( x )
lim
h
h0
Year
1984
1986
1988
1990
1992
1994
population 695
716
733
782
800
817
7. The population P (in thousands) of the city of San Jose, California, from 1984-1994 is given in the table
above.
a. Find the average rate of growth
(i) from 1986 to 1992
(ii) from 1988 to 1994
b. Estimate the instantaneous rate of growth in 1992.
2
8. The cost (in dollars) of producing x units of a certain commodity is C( x)  5000  10 x  0.05x .
a) Find the average rate of change of C with respect to x when the production level is changed
(i) from x = 100 to x = 105
(ii) from x = 100 to x = 101
b) Use the definition of derivative to find the instantaneous rate of change of C when x = 100. This is the
marginal cost! What does this mean? Does it agree with part (a)?
9. A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a
function of time. Is the initial rate of change greater or less than the rate of change after an hour?
10. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after t seconds is
2
given by H (t )  58t  0.83t .
a) Find the average velocity of the arrow over the first 2 seconds.
b) When will the arrow hit the moon?
c) Estimate the velocity when the arrow hits the moon.
d) When is the arrow at its maximum?
e) What is the maximum?
f) What is the velocity at the maximum?
***A particle is speeding up when the signs of the velocity and acceleration agree and the particle is slowing
down when the signs of the velocity and acceleration are opposite!****
11. The graph below shows the position function of a car. Use the shape of the graph to explain.
a) What was the initial velocity?
b) Was the car going faster at B or C? Why?
c) Was the car speeding up or slowing down at B, C & H?
d) What happened between D and E?
e) When is the car stopped?
12. A manufacturer produces toy airplanes (x). The cost C = f(x) is measured in dollars.
a) What is the meaning of f '(x) and what are the units?
b) What is the practical meaning of f ' (1000) = 9 ?
c) What is the meaning of f(300) = 1000?
d) Which do you think is greater f ' (50) or f ' (500)? Why?
e) What is f 1 (5000)  2000?
13. The tangent line to f(x) at (-2, 3) passes through (-1, 0). What is f ' (-2) and f(-2)?
14. Find the derivative of
f ( x)  3  2x  x2 at x = 2. Then write the equation of the tangent line at x = 2.
15. a) Using the table below, estimate the average rate of change on [3, 9].
b) Using the table, estimate f '(6) and f ' (9).
time t (min)
Area (sq. ft)
0
10
3
12
6
18
9
23
16. Sketch the graph of a function whose first and second derivatives are always negative.
17. Sketch the graph of a function whose first derivative is always negative and whose second derivative is
always positive.
18. The cost of living continues to rise, but as a slower rate. In terms of a function and its derivatives, what
does this mean? Sketch a graph to represent this situation.
19. Write the equation of the tangent line to y  x  5 at x = 4.
20. Use the definition of derivative to find f ' (x) when f ( x) 
3
x2
Answers to Review Sheet:
1. a) 3 km/min b) f ' (5)≈2
2. The instantaneous rate of change of a function at a specific x value—the slope of the
1
tangent line at a specific x value. 3. a) f ' (x) = 2x + 5 b) f ' (x) = 0 c) f ' (x) =  2
d) f (x)=
x3
2 x
4. i) c and e ii) d iii) b- cusp d—infinite discontinuity f-vert tangent iv)(a, b) and (c, d) v) (b, c) (d, e) (e,g)
vi)(b, d) (d, e) (f, g) vii) (a, b) (e,f)
5—see notes 6.
7ai) 14 ii) 14 b) 8.75 8a i) 20.25 ii) 20.05 b) f ' (x) = 10 + 0.1x f ' (100) = 20 The rate of change in the
cost of production is $20/hr when 100 units have been produced. It does appear the limit of the secant lines in
part a is approaching 20 as h approaches 0.
9.
greatest at the beginning
10. a) 56.34 meters/sec b) 69.880 sec c) -58 m/sec d)34.94 sec e) 1013.253 m f) 0 m/sec
11 a) 0 b) at B slope steeper c) B speeding up C slowing down H speeding up
d) he stopped at a light maybe e) A, [D, E], G, I
12 a) the change in cost $ with respect to the number of airplanes produced dollars/planes
b) When 1000 planes are produced, the cost to produce the next plane would be $9.00
c) When 300 planes are produced, the cost is $1000
d) f '(50) the more you produce, the less cost/item will be.
e) When cost to produce planes reaches $5000, the company should be producing 2000 planes.
13) f(-2) = 3 and f ' (-2) = -3. 14) f ' (x)= -2 -2x f '(2) = -6 tangent: y+5 = -6 (x-2)
15) f ' (6) ≈ 11/6 f ' (9) ≈ 5/3
16.
17.
18.
f inc so f ' + f conc down f″ -
19. y-3= (1/6)(x-4)
20. y  
3
( x  2) 2