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

2. Continuity & differentiability
0
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v3s² / 4

3. sin²x
u'/((ln a) u)
Limits
(1 - cos 2x) / 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).

4. Volume of a Sphere
1/2
Derivative of position at a point
v2/2
V = 4/3 pi r^3

5. d/dx[arcsec x]
1/(|x|v(x²-1))
pr²h/3
Derivative of Position - s'(t)
1

6. Continuity on an open interval - (a -b)
v3/2
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
S = 4 pi r^2
-1

7. Instantaneous velocity
1 / cos x
Derivative of position at a point
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

8. d/dx[e^x]
e^x
-csc x cot x
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
pr²

9. d/dx[cos x]
-1/(|x|v(x²-1))
e^x
f is an even function
-sin x

10. cot x
u'/u - u > 0
sin x / cos x
-sin x
1 / tan x = cos x / sin x

11. sin p/4
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'
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
0
v2/2

12. sin 0
(ln a) a^x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

13. How to get from precalculus to calculus
-csc x cot x
f(x) g'(x) + g(x) f'(x)
Limits
sec²x

14. cos p/2
csc²x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
-1
0

15. cos 3p/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
-1/(|x|v(x²-1))
1/(1+x²)
0

16. Alternate Limit Definition of a derivative

17. d/dx[ln u]

18. sin p/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
1/x - x>0
v3/2

19. d/dx[cot x]
Slope of a function at a point/slope of the tangent line to a function at a point
e^x
-csc² x
1/x - x>0

20. Area of an equilateral triangle
v3s² / 4
Limits
-1
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

21. sin p/3
v3/2
n ln m
1
1

22. d/dx[sin x]
cos x
d/dx[c] = 0
v3s² / 4
-csc² x

23. Average speed
u'/((ln a) u)
f(x) g'(x) + g(x) f'(x)
?s/?t
Derivative of Position - s'(t)

24. sec x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1 / cos x
1 / sin x

25. cos p/4
1/x - x>0
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'
v2/2

26. cos p
(1 + cos 2x) / 2
-1
(1 - cos 2x) / 2
V = 4/3 pi r^3

27. d/dx[ln x]
1/v(1-x²)
sec²x
1/x - x>0
d/dx[cf(x)] = c f'(x)

28. d/dx[log_a x]
u' (ln a) a^u
1/2
1/((ln a) x)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x

29. d/dx[x]
1
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
u'/((ln a) u)
1 / sin x

30. Constant Multiple Rule for Derivatives

31. csc x
-1/(1+x²)
f is an even function
f'(g(x))g'(x)
1 / sin x

32. sin p
pr²h/3
0
(ln a) a^x
-sin x

33. Indeterminate form
n ln m
1
0/0
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

34. sin p/6
Derivative of position at a point
1/2
-csc² x
d/dx[cf(x)] = c f'(x)

35. ln e
?s/?t
1
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
-1

36. ln 1
-csc² 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).
sec²x
0

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

38. Circumference of a circle
0
-1
2pr
e^x

39. Surface Area of a Sphere
(ln a) a^x
S = 4 pi r^2
0
-csc² x

40. Power Rule for Derivatives
d/dx[x^n]=nx^(n-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.
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
1 / tan x = cos x / sin x

41. d/dx[sec x]
-sin x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
u'/u - u > 0
sec x tan x

42. cos p/6
v3/2
0
sec x tan x
-1/v(1-x²)

43. d/dx[tan x]
-1/(1+x²)
u'/u - u > 0
sec² x
1

44. cos²x + sin²x
v3/2
Derivative of Position - s'(t)
1/2
1

45. The Quotient Rule

46. The limit as x approaches 0 of (1 - cos x) / x
0
-1/(1+x²)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
x values where f'(x) is zero or undefined.

47. Rolle's Theorem

48. The Product Rule

49. cos p/3
S = 4 pi r^2
1/2
n ln m
0/0

50. d/dx[arccot x]
Limits
pr²h
1 / cos x
-1/(1+x²)

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