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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. Guidelines for implicit differentiation

2. Area of a circle
-1/(|x|v(x²-1))
1
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
pr²

3. d/dx[e^u]

4. d/dx[arccot 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).
f is an even function
-1/(1+x²)
1/2

5. ln (m/n)
1 / sin 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'
1
ln m - ln n

6. sin p/4
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
v2/2
1 / cos x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

7. Instantaneous velocity
Derivative of position at a point
Derivative of Position - s'(t)
ln m + ln n
d/dx[c] = 0

8. Mean Value Theorem

9. d/dx[log_a x]
1 / tan x = cos x / sin x
0
-1/(1+x²)
1/((ln a) x)

10. ln (mn)
pr²h/3
x values where f'(x) is zero or undefined.
ln m + ln n
v2/2

11. Derivative of a constant
d/dx[c] = 0
-1
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

12. Volume of a Sphere
?s/?t
pr²h/3
V = 4/3 pi r^3
f'(g(x))g'(x)

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

14. Velocity - v(t)

15. d/dx[sec x]
sec x tan x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
cos x
d/dx[cf(x)] = c f'(x)

16. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
(ln a) a^x
-1/v(1-x²)
1

17. d/dx[ln u]

18. d/dx[log_a u]

19. If f(-x) = -f(x)
0
sec²x
f is an odd function
?s/?t

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

21. Circumference of a circle
csc²x
2 sin x cos x
2pr
1/x - x>0

22. Critical number

23. Indeterminate form
-csc x cot x
f is an odd function
0/0
v3/2

24. cos²x
v2/2
-1/(1+x²)
sec² x
(1 + cos 2x) / 2

25. Volume of a right circular cylinder
2 sin x cos x
pr²h
v2/2
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'

26. cos(2x)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
f(x) g'(x) + g(x) f'(x)
cos²x - sin²x
1 / cos x

27. How to get from precalculus to calculus
1/2
0/0
csc²x
Limits

28. d/dx[arccsc x]
ln m - ln n
f is an odd function
-1/(|x|v(x²-1))
Slope of a function at a point/slope of the tangent line to a function at a point

29. csc x
1 / sin x
-csc x cot x
(ln a) a^x
v3/2

30. The Product Rule

31. d/dx[cos x]
-1/(1+x²)
(1 + cos 2x) / 2
sin x / cos x
-sin x

32. cos 0
1
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
ln m - ln n
1/2

33. d/dx[a^u]

34. Continuity at a point (x = c)
1/((ln a) x)
-1
v3s² / 4
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)

35. Sum and Difference Rules for Derivatives

36. d/dx[x]
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1
u'/((ln a) u)
1

37. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
d/dx[x^n]=nx^(n-1)
1
-csc x cot x

38. Surface Area of a Sphere
1/((ln a) x)
f'(g(x))g'(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.
S = 4 pi r^2

39. d/dx[arctan x]
1/(1+x²)
1/(|x|v(x²-1))
0
1 / cos x

40. Average speed
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)
cos²x - sin²x
0
?s/?t

41. sin p
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)
0
1/((ln a) x)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

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

43. Continuity on an open interval - (a -b)
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
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
1 / sin x
u' (ln a) a^u

44. ln e
1
f is an even function
pr²h/3
n ln m

45. Guidelines for solving related rates problems
(1 - cos 2x) / 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
u'/((ln a) u)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

46. ln 1
0
f is an odd function
cos x
d/dx[x^n]=nx^(n-1)

47. d/dx[sin x]
-csc² x
u'/((ln a) u)
-csc x cot x
cos x

48. cot x
1 / tan x = cos x / sin x
ln m - ln n
Limits
f'(g(x))g'(x)

49. Continuity on a closed interval - [a -b]
1/x - x>0
1
-csc x cot x
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)

50. Constant Multiple Rule for Derivatives