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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[arcsin 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)
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).
1/v(1-x²)
d/dx[c] = 0

2. Limit Definition of a Derivative

3. sin p/3
v3/2
sec²x
1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

4. Average speed
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)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
?s/?t
1/x - x>0

5. d/dx[arcsec x]
1/(|x|v(x²-1))
d/dx[cf(x)] = c f'(x)
1/x - x>0
-1/(|x|v(x²-1))

6. cot x
x values where f'(x) is zero or undefined.
f is an even function
f(x) g'(x) + g(x) f'(x)
1 / tan x = cos x / sin x

7. d/dx[arctan x]
u'/u - u > 0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
0/0
1/(1+x²)

8. d/dx[cos x]
1
0
u'/((ln a) u)
-sin x

9. sec x
1 / cos x
f(x) g'(x) + g(x) f'(x)
pr²h
2 sin x cos x

10. sin p/2
v2/2
1
v3s² / 4
Slope of a function at a point/slope of the tangent line to a function at a point

11. cos p/2
cos x
0
Derivative of Position - s'(t)
f'(g(x))g'(x)

12. cos 0
ln m + ln n
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1
d/dx[cf(x)] = c f'(x)

13. d/dx[sec x]
sec x tan x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
d/dx[cf(x)] = c f'(x)
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

14. 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.
Slope of a function at a point/slope of the tangent line to a function at a point
f(x) g'(x) + g(x) f'(x)
0

15. Velocity - v(t)

16. sin(2x)
2 sin x cos x
2pr
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v3/2

17. Continuity & differentiability
v3/2
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-1/(1+x²)

18. Sum and Difference Rules for Derivatives

19. sin p
-1
f(x) g'(x) + g(x) f'(x)
0
v2/2

20. d/dx[arccos x]
2pr
0
0
-1/v(1-x²)

21. d/dx[e^u]

22. sin p/4
cos²x - sin²x
Limits
v2/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

23. d/dx[log_a x]
1/((ln a) x)
-1/(1+x²)
ln m + ln n
v3/2

24. cos p/3
d/dx[x^n]=nx^(n-1)
1/2
0/0
-1

25. Indeterminate form
0
0/0
d/dx[x^n]=nx^(n-1)
d/dx[c] = 0

26. d/dx[log_a u]

27. Area of a circle
pr²
v3s² / 4
1 / cos x
-1

28. d/dx[a^x]
f is an odd function
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
(ln a) a^x

29. tan x
v3/2
sin x / cos x
-1/v(1-x²)
1/(1+x²)

30. How to get from precalculus to calculus
f is an even function
Derivative of Position - s'(t)
Limits
?s/?t

31. Surface Area of a Sphere
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
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
S = 4 pi r^2
u'/((ln a) u)

32. d/dx[arccsc x]
-1/v(1-x²)
1 / cos x
-1/(|x|v(x²-1))
v2/2

33. ln e
ln m - ln n
1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u' e^u

34. 1 + tan²x
?s/?t
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)
sec²x
csc²x

35. ln 1
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f is an odd function
Slope of a function at a point/slope of the tangent line to a function at a point
0

36. cos p
u'/u - u > 0
cos x
-1
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)

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

38. d/dx[tan x]
cos x
?s/?t
f is an even function
sec² x

39. Volume of a cone
v3/2
0
-1/v(1-x²)
pr²h/3

40. Derivative of a constant
-csc² x
d/dx[c] = 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1/(|x|v(x²-1))

41. 1 + cot²x
csc²x
1/((ln a) x)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
u'/((ln a) u)

42. cos(2x)
v3/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
cos²x - sin²x
Derivative of Position - s'(t)

43. sin 0
1
v2/2
0
u' (ln a) a^u

44. cos²x + sin²x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1
e^x

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

46. Derivative
1
f(x) g'(x) + g(x) f'(x)
Slope of a function at a point/slope of the tangent line to a function at a point
sec²x

47. Rolle's Theorem

48. Guidelines for solving related rates problems
-csc x cot x
u'/((ln a) u)
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

49. ln (m/n)
x values where f'(x) is zero or undefined.
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
n ln m
ln m - ln n

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