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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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[x]
1
1/(|x|v(x²-1))
pr²
1 / tan x = cos x / sin x

2. How to get from precalculus to calculus
Limits
v3/2
1/(1+x²)
-csc x cot x

3. d/dx[arccos x]
1/v(1-x²)
1 / tan x = cos x / sin x
2pr
-1/v(1-x²)

4. sin p/3
v3/2
1
(1 + cos 2x) / 2
S = 4 pi r^2

5. Position function of a falling object (with acceleration in ft/s²)
e^x
u' e^u
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/x - x>0

6. Critical number

7. Derivative
d/dx[x^n]=nx^(n-1)
Slope of a function at a point/slope of the tangent line to a function at a point
-1/(|x|v(x²-1))
pr²h/3

8. d/dx[arccot x]
0
-1/(1+x²)
f'(g(x))g'(x)
f(x) g'(x) + g(x) f'(x)

9. d/dx[a^u]

10. d/dx[a^x]
Slope of a function at a point/slope of the tangent line to a function at a point
(ln a) a^x
0/0
sec x tan x

11. csc x
1 / sin x
-csc² x
(ln a) a^x
0

12. cos p/6
1
v3/2
u'/u - u > 0
1/2

13. Volume of a right circular cylinder
1
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
pr²h

14. cos²x
(1 + cos 2x) / 2
(1 - cos 2x) / 2
sec x tan x
u'/((ln a) u)

15. d/dx[e^x]
Slope of a function at a point/slope of the tangent line to a function at a point
-1/(|x|v(x²-1))
sec²x
e^x

16. If f(-x) = -f(x)
d/dx[cf(x)] = c f'(x)
f is an odd function
ln m - ln n
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).

17. d/dx[ln x]
V = 4/3 pi r^3
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/v(1-x²)
1/x - x>0

18. cos p/2
-1/v(1-x²)
0
Derivative of position at a point
sec² x

19. d/dx[log_a x]
1 / tan x = cos x / sin x
1/((ln a) x)
Slope of a function at a point/slope of the tangent line to a function at a point
?s/?t

20. ln mn
n ln m
1 / sin x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
-1

21. Continuity on an open interval - (a -b)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
u' e^u
1

22. sin p/6
1/2
0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1

23. d/dx[arctan x]
u'/u - u > 0
0
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/(1+x²)

24. 1 + tan²x
-csc² x
cos²x - sin²x
f is an odd function
sec²x

25. Indeterminate form
1/2
Slope of a function at a point/slope of the tangent line to a function at a point
2pr
0/0

26. d/dx[ln u]

27. d/dx[cot x]
-csc² x
1/x - x>0
-1/(1+x²)
0

28. Velocity - v(t)

29. Rolle's Theorem

30. d/dx[arcsin x]
1/v(1-x²)
1
-1
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'

31. Mean Value Theorem

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

33. cos p/4
ln m - ln n
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.
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

34. cos 3p/2
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1
0

35. The Product Rule

36. d/dx[sin x]
cos x
sin x / cos x
1
0

37. Circumference of a circle
2pr
0
sec² x
S = 4 pi r^2

38. Volume of a Sphere
S = 4 pi r^2
d/dx[c] = 0
V = 4/3 pi r^3
d/dx[cf(x)] = c f'(x)

39. ln (mn)
ln m + ln n
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x

40. Extreme Value Theorem
2 sin x cos x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
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).
(ln a) a^x

41. sin p/4
1/(1+x²)
S = 4 pi r^2
v2/2
Slope of a function at a point/slope of the tangent line to a function at a point

42. Power Rule for Derivatives
ln m + ln n
u' e^u
d/dx[x^n]=nx^(n-1)
-1/v(1-x²)

43. Sum and Difference Rules for Derivatives

44. d/dx[arccsc x]
-1/(|x|v(x²-1))
1/2
0/0
-1/(1+x²)

45. ln e
u'/u - u > 0
0
1
pr²h

46. sin²x
0
u' (ln a) a^u
f(x) g'(x) + g(x) f'(x)
(1 - cos 2x) / 2

47. tan x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
sin x / cos x
ln m - ln n
1

48. sin p
S = 4 pi r^2
sec²x
pr²h/3
0

49. sin 3p/2
csc²x
-1
1 / tan x = cos x / sin x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

50. cos(2x)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v2/2
S = 4 pi r^2
cos²x - sin²x