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AP Calculus Formulas

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. Average speed
?s/?t
(ln a) a^x
0
d/dx[cf(x)] = c f'(x)

2. d/dx[a^x]
-csc² x
(ln a) a^x
1/v(1-x²)
f is an odd function

3. d/dx[tan x]
f is an odd function
sec² x
0
cos²x - sin²x

4. sec x
1. Differentiate both sides w.r.t. x 2. Move all y' terms to one side & other terms to the other 3. Factor out y' 4. Divide to solve for y'
sec x tan x
1 / cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

5. d/dx[x]
f(x) g'(x) + g(x) f'(x)
1
V = 4/3 pi r^3
-1/v(1-x²)

6. Volume of a cone
0
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.
x values where f'(x) is zero or undefined.
pr²h/3

7. sin(2x)
pr²h
2 sin x cos x
n ln m
pr²h/3

8. Rolle's Theorem

9. d/dx[ f(x) / g(x) ]

10. d/dx[csc x]
v3s² / 4
(1 - cos 2x) / 2
-csc x cot x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

11. sin p/3
v3/2
1/((ln a) x)
0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

12. Continuity on a closed interval - [a -b]
1/(1+x²)
1. f(x) is continuous on the closed interval (a -b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)
u' (ln a) a^u
cos²x - sin²x

13. d/dx[e^x]
1. Given - Want - Sketch 2. Write an equation using variables given/to be determined 3. Differentiate w.r.t. time (using chain rule) 4. Plug in & solve
e^x
pr²h/3
sin x / cos x

14. The Quotient Rule

15. 1 + cot²x
S = 4 pi r^2
u'/((ln a) u)
0
csc²x

16. sin 3p/2
S = 4 pi r^2
-1
0/0
if f(x) is continuous and differentiable - slope of tangent line equals slope of secant line at least once in the interval (a - b) - f '(c) = [f(b) - f(a)]/(b - a)

17. cos p
1
v3/2
-1
x values where f'(x) is zero or undefined.

18. Intermediate Value Theorem
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1
if f(x) is continuous and differentiable - slope of tangent line equals slope of secant line at least once in the interval (a - b) - f '(c) = [f(b) - f(a)]/(b - a)

19. Circumference of a circle
-sin x
cos²x - sin²x
V = 4/3 pi r^3
2pr

20. ln mn
Limits
d/dx[c] = 0
n ln m
-1/(1+x²)

21. 1 + tan²x
1/2
1/2
sec²x
cos x

22. d/dx[arccsc x]
ln m - ln n
-1/v(1-x²)
-1/(|x|v(x²-1))
1/((ln a) x)

23. d/dx[arccos x]
sin x / cos x
pr²
-1/v(1-x²)
u'/((ln a) u)

24. Critical number

25. Guidelines for solving related rates problems
1. f(x) is defined at f(c) 2. The limit as x approaches c of f(x) exists 3. The limit as x approaches c of f(x) = f(c)
f(x) g'(x) + g(x) f'(x)
1. Given - Want - Sketch 2. Write an equation using variables given/to be determined 3. Differentiate w.r.t. time (using chain rule) 4. Plug in & solve
u' e^u

26. Constant Multiple Rule for Derivatives

27. Derivative of a constant
d/dx[c] = 0
sec²x
0/0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

28. ln (mn)
1/2
f is an even function
ln m + ln n
u'/((ln a) u)

29. Volume of a Sphere
-1
pr²h/3
V = 4/3 pi r^3
pr²h

30. Volume of a right circular cylinder
pr²h
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1
Let f be continuous on [a -b] and differentiable on (a -b) and if f(a)=f(b) then there is at least one number c on (a -b) such that f'(c)=0 (If the slope of the secant is 0 - the derivative must = 0 somewhere in the interval).

31. cos p/2
1
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
0
1

32. d/dx[log_a x]
0/0
v2/2
1
1/((ln a) x)

33. ln e
1
-1/(1+x²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1 / sin x

34. If f(-x) = -f(x)
f is an odd function
1/2
ln m - ln n
2pr

35. cos p/6
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
f'(g(x))g'(x)
v3/2
V = 4/3 pi r^3

36. Power Rule for Derivatives
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
V = 4/3 pi r^3
pr²
d/dx[x^n]=nx^(n-1)

37. How to get from precalculus to calculus
Limits
u' e^u
V = 4/3 pi r^3
0

38. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1 / tan x = cos x / sin x
f'(g(x))g'(x)
d/dx[c] = 0

39. sin p
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
2 sin x cos x
cos x
0

40. d/dx[e^u]

41. Chain Rule: d/dx[f(g(x))] =

42. tan x
1
sin x / cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
Let f be continuous on [a -b] and differentiable on (a -b) and if f(a)=f(b) then there is at least one number c on (a -b) such that f'(c)=0 (If the slope of the secant is 0 - the derivative must = 0 somewhere in the interval).

43. The limit as x approaches 0 of sin x / x
1
pr²h/3
1. f(x) is continuous on the closed interval (a -b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)
v3/2

44. d/dx[arctan x]
1/(1+x²)
-csc x cot x
1
csc²x

45. ln 1
S = 4 pi r^2
-1/(1+x²)
0
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

46. sin 0
sin x / cos x
0/0
-1/v(1-x²)
0

47. cos p/3
u'/u - u > 0
(1 - cos 2x) / 2
1/2
pr²h

48. Velocity - v(t)

49. d/dx[arcsin x]
1/((ln a) x)
1/v(1-x²)
(ln a) a^x
1. Differentiate both sides w.r.t. x 2. Move all y' terms to one side & other terms to the other 3. Factor out y' 4. Divide to solve for y'

50. Mean Value Theorem