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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. Power Rule for Derivatives
v3/2
cos²x - sin²x
d/dx[x^n]=nx^(n-1)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

2. d/dx[arccos x]
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 / cos x
-1/v(1-x²)
f'(g(x))g'(x)

3. Area of an equilateral triangle
v3s² / 4
1
1
ln m - ln n

4. The Product Rule

5. The limit as x approaches 0 of (1 - cos x) / x
0
Derivative of position at a point
f'(g(x))g'(x)
v2/2

6. sin²x
x values where f'(x) is zero or undefined.
0
(1 - cos 2x) / 2
cos²x - sin²x

7. d/dx[arccot x]
-1/(|x|v(x²-1))
-1/(1+x²)
v3/2
-sin x

8. cos²x + sin²x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
0
1
cos x

9. Position function of a falling object (with acceleration in m/s²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-sin x
2pr

10. d/dx[a^u]

11. d/dx[tan x]
(1 - cos 2x) / 2
pr²h
Limits
sec² x

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

13. Position function of a falling object (with acceleration in ft/s²)
0
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/2
1/v(1-x²)

14. Sum and Difference Rules for Derivatives

15. d/dx[ln u]

16. d/dx[log_a x]
1/((ln a) x)
-1
u'/u - u > 0
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

17. Volume of a cone
V = 4/3 pi r^3
pr²h/3
pr²h
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)

18. tan x
V = 4/3 pi r^3
d/dx[cf(x)] = c f'(x)
e^x
sin x / cos x

19. d/dx[e^u]

20. Extreme Value Theorem
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
-1/(|x|v(x²-1))
1/2
cos²x - sin²x

21. sin p/3
1 / cos x
Limits
sec² x
v3/2

22. d/dx[arccsc x]
d/dx[c] = 0
-1/(|x|v(x²-1))
v3/2
-1

23. ln e
v3s² / 4
1/2
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

24. Continuity at a point (x = c)
-csc² x
e^x
V = 4/3 pi r^3
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)

25. ln (m/n)
-1
1
0
ln m - ln n

26. sin 3p/2
Derivative of Position - s'(t)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1
1 / sin x

27. Derivative of an inverse (if g(x) is the inverse of f(x))

28. Mean Value Theorem

29. Volume of a Sphere
V = 4/3 pi r^3
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'
1/x - x>0
u'/((ln a) u)

30. ln mn
n ln m
S = 4 pi r^2
f is an odd function
f(x) g'(x) + g(x) f'(x)

31. cot x
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 / sin x
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 / tan x = cos x / sin x

32. ln (mn)
v3s² / 4
ln m + ln n
u' e^u
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

33. Constant Multiple Rule for Derivatives

34. cos(2x)
1/2
f is an odd function
cos²x - sin²x
Derivative of position at a point

35. Continuity on an open interval - (a -b)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
sec²x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
Limits

36. Indeterminate form
1/x - x>0
0/0
-csc x cot 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'

37. sec x
-1/(|x|v(x²-1))
pr²h/3
1
1 / cos x

38. Intermediate Value Theorem
1/((ln a) x)
S = 4 pi r^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.
f'(g(x))g'(x)

39. sin p/6
1/2
v2/2
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

40. cos²x
0/0
(1 + cos 2x) / 2
?s/?t
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

41. If f(-x) = -f(x)
-1/v(1-x²)
f is an odd function
S = 4 pi r^2
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

42. Average speed
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
v2/2
?s/?t
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

43. sin p/4
Limits
d/dx[c] = 0
v2/2
1

44. d/dx[cos x]
1 / sin x
Derivative of Position - s'(t)
-sin x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

45. 1 + cot²x
1/(|x|v(x²-1))
csc²x
d/dx[cf(x)] = c f'(x)
1/((ln a) x)

46. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
v2/2
1
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

47. cos p/6
pr²h/3
Slope of a function at a point/slope of the tangent line to a function at a point
v3/2
1

48. d/dx[arcsec x]
sec²x
0
1/(|x|v(x²-1))
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

49. The Quotient Rule

50. csc x
1/2
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
2 sin x cos x
1 / sin x