CHapter 3 Memory Items
Short Answer
1. What is the slope of the secant line connecting (a,f(a)) and (b,f(b)) ?
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2. What is the average rate of change of f(x) on [ a , b ] ?
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3. What is the definition of the slope of the tangent line at x ?
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4. What is the definition of f ' (x) ?
.
5. What is the definition of the instantaneous rate of change in f(x) ?
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6. Give two or three other phrases that mean the same thing as the derivative of a function.
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7. How is the existence or non existence of f ' (c) related to the continuity of f at x = c ?
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8. If f(x) is continuous at x = c, then what do we know about the derivative of f at x = c ?
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9. If f(x) has a sharp corner at x = c, what do we know about f ' (c) ?
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10. If the derivative does not exist at x = c but it's limit is infinite as x
must occur at x = c on the curve?
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11. d/dx (c) = ____________________
12. d/dx (mx + b) = _______________
13. d/dx ( mx ) = _______________
14. d/dx ( x ) = ________________________
15. d/dx ( f(x) + g(x) ) = __________________________________
16. What is the Constant Multiplier Rule ?
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17. What is the Product Rule?
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18. What is the Quotient Rule?
c, and f(x) is continuous at x = c, what
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19. d/dx ( sin x ) = ____________________
20. d/dx ( cos x ) = ___________________________
21. d/dx ( tan x ) = ________________________________________
22. d/dx ( sec x ) = _______________________________________
23. d/dx (cot x ) = __________________________________________
24. d/dx (csc x) = ________________________________________________
25. What is the Chain Rule (Newton's notation)
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26. What is the Chain Rule (Leibniz's notation)
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27. Given dy/dx, what is the independent variable?
28. Given dy/dx, what is the dependent variable?
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29. Given du/dt, what is the independent variable?
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30. Given du/dt, what is the dependent variable?
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31. If x(t) gives position, give two expressions for velocity.
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32. If x(t) gives position, give three expressions for acceleration.
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33. When is a particle at rest?
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34. When is a particle speeding up?
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35. When is a particle slowing down?
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36. If v(t) > 0 then the particle is ____________________________________________.
37. If v(t) < 0 then the particle is ____________________________________________.
38. What is the expression for the speed of a particle whose position is x(t) ?
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39. What is a normal line ?
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40. d/dx (f(y)) = ________________________
41. What is ALWAYS the independent variable in related rates problems?
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42. Why is it important to use variable names for all variable quantities until after the derivative is taken in a
related rates problem?
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43. In a related rates problem, the derivative of t is ______________________.
44. In a related rates problem, the derivative of x is ______________________.
45. In a related rates problem, the derivative of y is ______________________.
46. In a related rates problem, the derivative of V is ______________________.
47. In a related rates problem, the derivative of A is ______________________.
48. What is the expression for the "error" created when approximating a y-value by using a tangent line drawn
from a neighboring point?
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CHapter 3 Memory Items
Answer Section
SHORT ANSWER
1. (f(b) - f(a)) / (b - a)
2. (f(b) - f(a)) / (b - a)
3. lim f(x+h) - f(x)
or
lim f(x) - f(t)
h 0
h
t x
x-t
4. lim f(x+h) - f(x)
or
lim f(x) - f(t)
h 0
h
t x
x-t
5. lim f(x+h) - f(x)
or
lim f(x) - f(t)
h 0
h
t x
x-t
6. instantaneous rate of change, slope of the tangent, and/or slope of the curve
7. If f ' (c) exists the f is continuous at x = c. If f '(c) DNE then f is either discontinuous, vertical, or has a sharp
corner at x = c.
8. f ' (c) might exist as long as there is not a sharp corner.
9. It does not exist.
10. a vertical tangent line
11. 0
12. m
13. m
14.
15.
16.
17.
18.
nx
f ' (x) + g ' (x)
d/dx (c f(x) ) = c f ' (x)
d/dx (f(x) · g(x)) = f(x) · g ' (x) + f ' (x) · g(x)
d/dx ( f(x) / g(x) ) = g(x) · f ' (x) - g ' (x) · f(x)
(g(x))
19. cos x
20. - sin x
21. sec x
22. sec x tan x
23.
24.
25.
26.
27.
28.
29.
30.
- csc x
- csc x cot x
d/dx [f(g(x))] = f ' (g(x)) · g ' (x)
dy du
dy
--- · --- = ---du dx
dx
x
y
t
u
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
x ' (t) and v(t)
x '' (t) , v ' (t) , and a(t)
when v(t) = 0
when v(t) and a(t) are either both negative or both positive
When v(t) and a(t) have opposite signs.
moving right or up
moving left or down
| x ' (t) | or | v(t) |
perpendicular to the tangent line
f ' (y) · dy/dx
t
Otherwise the derivatives will be zero (useless)
1
dx/dt
dy/dt
dV/dt
dA/dt
h · f ' (x) or x · f ' (x)