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AP Calculus Formulas

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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. Limit Definition of a Derivative

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

3. 1 + tan²x
sec²x
?s/?t
ln m - ln n
-1

4. cos p
-1
u' (ln a) a^u
1/2
-sin x

5. ln (mn)
csc²x
1
pr²h/3
ln m + ln n

6. sin 3p/2
-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'
1/(|x|v(x²-1))
0

7. d/dx[cos x]
-sin x
1/(1+x²)
u' e^u
(1 - cos 2x) / 2

8. cos p/6
cos²x - sin²x
1
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.
v3/2

9. d/dx[tan x]
pr²
sec² x
1/(1+x²)
d/dx[x^n]=nx^(n-1)

10. d/dx[e^x]
1/v(1-x²)
Derivative of Position - s'(t)
e^x
-1/v(1-x²)

11. ln 1
csc²x
0
u' e^u
V = 4/3 pi r^3

12. sin p
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
0
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

13. d/dx[log_a u]

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

15. Velocity - v(t)

16. d/dx[arccsc 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).
-sin x
1/(|x|v(x²-1))
-1/(|x|v(x²-1))

17. Sum and Difference Rules for Derivatives

18. Continuity on a closed interval - [a -b]
1 / cos x
-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)
Limits

19. If f(-x) = f(x)
n ln m
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
f is an even function
u'/u - u > 0

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

21. d/dx[x]
0
Derivative of Position - s'(t)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1

22. Indeterminate form
0/0
S = 4 pi r^2
-1/(|x|v(x²-1))
Limits

23. Continuity at a point (x = c)
Derivative of Position - s'(t)
-csc x cot x
-1/v(1-x²)
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)

24. cos p/2
0
f'(g(x))g'(x)
pr²h/3
-1/(|x|v(x²-1))

25. Position function of a falling object (with acceleration in ft/s²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v2/2
v3s² / 4
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

26. Continuity on an open interval - (a -b)
Slope of a function at a point/slope of the tangent line to a function at a point
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
sec²x
-1

27. Guidelines for implicit differentiation

28. sin²x
Limits
(1 - cos 2x) / 2
1/2
(ln a) a^x

29. sin p/4
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.
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/2
v2/2

30. sin p/6
1/((ln a) x)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
1/2
v3/2

31. Surface Area of a Sphere
0
S = 4 pi r^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).
x values where f'(x) is zero or undefined.

32. sin p/3
v3/2
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
V = 4/3 pi r^3
1/v(1-x²)

33. d/dx[a^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).
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
(ln a) a^x
0/0

34. d/dx[arccot x]
-1/(1+x²)
sec²x
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)
1/x - x>0

35. sin(2x)
2 sin x cos x
1/(|x|v(x²-1))
-1/(1+x²)
v3/2

36. Mean Value Theorem

37. Extreme Value Theorem
ln m - ln n
u'/u - u > 0
1 / tan x = cos x / sin x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

38. If f(-x) = -f(x)
0
f is an odd function
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.
0

39. Volume of a Sphere
2pr
n ln m
V = 4/3 pi r^3
(1 - cos 2x) / 2

40. How to get from precalculus to calculus
(1 + cos 2x) / 2
Limits
1
?s/?t

41. Intermediate Value Theorem
1/(|x|v(x²-1))
pr²h
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.

42. d/dx[sec x]
f is an odd function
sec x tan x
pr²h
x values where f'(x) is zero or undefined.

43. 1 + cot²x
csc²x
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
1 / cos x

44. cos 0
1
f is an even function
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)
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'

45. ln e
ln m - ln n
cos x
f is an even function
1

46. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
d/dx[cf(x)] = c f'(x)
0
u' e^u

47. cos(2x)
f is an even function
?s/?t
0/0
cos²x - sin²x

48. cos²x + sin²x
pr²
1
ln m - ln n
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)

49. d/dx[a^u]

50. d/dx[ln x]
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
0
1/2
1/x - x>0