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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. d/dx[arccsc x]
-1/(|x|v(x²-1))
0/0
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).
-csc x cot x

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

3. cos 0
f'(g(x))g'(x)
V = 4/3 pi r^3
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1

4. d/dx[csc x]
u'/u - u > 0
1/2
0
-csc x cot x

5. d/dx[arccos 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'
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/x - x>0

6. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/(|x|v(x²-1))
v3/2
v2/2

7. cos 3p/2
-1/v(1-x²)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
sec² x
0

8. The Quotient Rule

9. Area of an equilateral triangle
v3s² / 4
?s/?t
cos x
0

10. sin p/2
u'/((ln a) u)
f'(g(x))g'(x)
sec² x
1

11. d/dx[sin x]
v2/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-sin x
cos x

12. d/dx[cot x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
0
-csc² x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

13. Area of a circle
1/2
Slope of a function at a point/slope of the tangent line to a function at a point
pr²
(1 - cos 2x) / 2

14. sin p/4
S = 4 pi r^2
v2/2
0
-sin x

15. d/dx[arccot x]
0
-1/(1+x²)
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).
v2/2

16. If f(-x) = f(x)
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
sec²x

17. 1 + tan²x
(1 - cos 2x) / 2
sec²x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
ln m - ln n

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

19. sin p
0
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
1/(1+x²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

20. Derivative of a constant
v3s² / 4
Slope of a function at a point/slope of the tangent line to a function at a point
sec x tan x
d/dx[c] = 0

21. Volume of a cone
pr²h/3
1/(|x|v(x²-1))
f(x) g'(x) + g(x) f'(x)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

22. cot x
ln m - ln n
1 / tan x = cos x / sin x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1

23. d/dx[x]
0
1
e^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)

24. Mean Value Theorem

25. ln (m/n)
f'(g(x))g'(x)
ln m - ln n
v3/2
u' e^u

26. d/dx[e^u]

27. Sum and Difference Rules for Derivatives

28. Volume of a Sphere
v2/2
sec² x
0
V = 4/3 pi r^3

29. d/dx[sec x]
0
d/dx[x^n]=nx^(n-1)
sec x tan x
f is an odd function

30. cos p/2
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)
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

31. Instantaneous velocity
?s/?t
0
1 / sin x
Derivative of position at a point

32. How to get from precalculus to calculus
V = 4/3 pi r^3
1
Limits
1 / sin x

33. d/dx[log_a u]

34. d/dx[ln x]
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u'/u - u > 0
1/x - x>0
-1/v(1-x²)

35. Extreme Value Theorem
cos x
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].
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)

36. cos p/3
0
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1/2

37. Circumference of a circle
-csc x cot x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
2pr
1 / cos x

38. Constant Multiple Rule for Derivatives

39. d/dx[a^u]

40. Continuity on a closed interval - [a -b]
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/v(1-x²)
0/0
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)

41. 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.
v3/2
0
cos x

42. cos p
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1
ln m - ln n
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

43. Position function of a falling object (with acceleration in m/s²)
v3/2
1/((ln a) x)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m + ln n

44. Volume of a right circular cylinder
sec x tan x
1
pr²h
cos x

45. Alternate Limit Definition of a derivative

46. Surface Area of a Sphere
-1
S = 4 pi r^2
V = 4/3 pi r^3
sec x tan x

47. sin 3p/2
u' e^u
-1
1
0/0

48. ln 1
n ln m
0
f is an odd function
Slope of a function at a point/slope of the tangent line to a function at a point

49. d/dx[e^x]
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
(ln a) a^x
e^x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

50. d/dx[arcsin x]
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.
pr²h/3
1/v(1-x²)