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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. d/dx[e^x]
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
sec² x
e^x

2. d/dx[arccot x]
-1/(1+x²)
pr²
pr²h/3
sec²x

3. Sum and Difference Rules for Derivatives

4. Critical number

5. ln 1
1/v(1-x²)
-1
0
If two functions - f and g - are differentiable - then d/dx[ f(x) / g(x) ] = [g(x)f'(x) - f(x) g'(x)] / [g(x)]²

6. sin p/3
v3/2
ln m - ln n
1/((ln a) x)
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)

7. Alternate Limit Definition of a derivative

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

9. The limit as x approaches 0 of sin x / x
1
S = 4 pi r^2
Derivative of Position - s'(t)
v2/2

10. Constant Multiple Rule for Derivatives

11. cos 0
1
pr²h
-1
1 / tan x = cos x / sin x

12. Instantaneous velocity
Derivative of position at a point
0
1 / cos x
v3/2

13. Extreme Value Theorem
0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
1

14. cos 3p/2
(1 - cos 2x) / 2
0
pr²h
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

15. cos²x
sec x tan x
ln m + ln n
(1 + cos 2x) / 2
sec²x

16. ln mn
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)
sin x / cos x
n ln m
(ln a) a^x

17. d/dx[a^u]

18. Limit Definition of a Derivative

19. d/dx[e^u]

20. sin p/4
-1/v(1-x²)
v2/2
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.
1

21. 1 + tan²x
-csc x cot x
sec²x
v3/2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

22. Rolle's Theorem

23. ln (mn)
0
ln m + ln n
pr²h
1

24. sin p
0
1 / sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-csc² x

25. d/dx[sec x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
sec x tan x
n ln m
-1

26. Continuity on a closed interval - [a -b]
csc²x
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)
pr²
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

27. cos p/4
v2/2
sec²x
-1
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

28. cos p
Derivative of position at a point
-1
1
0

29. sec x
S = 4 pi r^2
1/x - x>0
1 / cos x
(1 - cos 2x) / 2

30. cos p/2
1/(|x|v(x²-1))
0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
d/dx[cf(x)] = c f'(x)

31. Continuity on an open interval - (a -b)
0/0
-1/v(1-x²)
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) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

32. The limit as x approaches 0 of (1 - cos x) / x
csc²x
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
0
-csc x cot x

33. Average speed
e^x
?s/?t
ln m - ln n
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

34. d/dx[csc x]
1 / tan x = cos x / sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-csc x cot x
e^x

35. Guidelines for implicit differentiation

36. If f(-x) = -f(x)
u' (ln a) a^u
1 / tan x = cos x / sin x
f is an odd function
1

37. Indeterminate form
Derivative of Position - s'(t)
e^x
0/0
f'(g(x))g'(x)

38. Intermediate Value Theorem
d/dx[cf(x)] = c f'(x)
sec x tan x
u' e^u
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.

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

40. Area of a circle
pr²
V = 4/3 pi r^3
e^x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

41. Area of an equilateral triangle
v3s² / 4
Derivative of Position - s'(t)
0
e^x

42. Continuity at a point (x = c)
1
2 sin x cos x
x values where f'(x) is zero or undefined.
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)

43. sin(2x)
2 sin x cos x
f'(g(x))g'(x)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
csc²x

44. How to get from precalculus to calculus
2 sin x cos x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
-1
Limits

45. d/dx[ln u]

46. d/dx[arccsc x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
2pr
-1/(|x|v(x²-1))
(1 - cos 2x) / 2

47. cot x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1 / tan x = cos x / sin x
u'/((ln a) u)
Derivative of position at a point

48. cos p/3
f(x) g'(x) + g(x) f'(x)
e^x
1/2
0

49. d/dx[log_a u]

50. d/dx[arcsec x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
-1
0/0
1/(|x|v(x²-1))