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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[log_a u]

2. Circumference of a circle
1
2pr
Derivative of Position - s'(t)
cos x

3. csc x
1 / sin x
V = 4/3 pi r^3
?s/?t
1/x - x>0

4. ln e
n ln m
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
S = 4 pi r^2
1

5. If f(-x) = -f(x)
1 / tan x = cos x / sin x
v2/2
f is an odd function
f(x) g'(x) + g(x) f'(x)

6. Area of an equilateral triangle
pr²
v3s² / 4
d/dx[cf(x)] = c f'(x)
u'/((ln a) u)

7. cos p/6
pr²h
f is an even function
v3/2
0

8. tan x
sin x / cos x
pr²h
1
csc²x

9. cos p/3
0/0
cos x
-1/(1+x²)
1/2

10. d/dx[arccos x]
-1/v(1-x²)
v3/2
d/dx[x^n]=nx^(n-1)
-csc x cot x

11. d/dx[e^u]

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

13. cos p/4
v2/2
1 / sin x
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).

14. Sum and Difference Rules for Derivatives

15. d/dx[a^x]
(ln a) a^x
(1 + cos 2x) / 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
cos x

16. d/dx[log_a x]
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/((ln a) x)
f is an odd function

17. Derivative of a constant
d/dx[c] = 0
-csc x cot x
1
0

18. Position function of a falling object (with acceleration in m/s²)
cos²x - sin²x
-1/v(1-x²)
d/dx[x^n]=nx^(n-1)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

19. cos p/2
x values where f'(x) is zero or undefined.
0
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
V = 4/3 pi r^3

20. Instantaneous velocity
1/v(1-x²)
Derivative of position at a point
1
cos x

21. The Product Rule

22. cos p
-1
v2/2
x values where f'(x) is zero or undefined.
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

23. 1 + tan²x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
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'
csc²x
sec²x

24. d/dx[arccsc x]
d/dx[x^n]=nx^(n-1)
ln m + ln n
-1/(|x|v(x²-1))
1/x - x>0

25. Limit Definition of a Derivative

26. If f(-x) = f(x)
cos x
pr²h
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 is an even function

27. cos(2x)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1
cos²x - sin²x
(1 - cos 2x) / 2

28. Constant Multiple Rule for Derivatives

29. d/dx[arcsec x]
1
1/(|x|v(x²-1))
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
-1

30. Alternate Limit Definition of a derivative

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

32. Intermediate Value Theorem
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/v(1-x²)
d/dx[x^n]=nx^(n-1)
1

33. d/dx[tan x]
d/dx[cf(x)] = c f'(x)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
pr²h/3
sec² x

34. Continuity at a point (x = c)
cos x
n ln m
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)
sec²x

35. cos 0
-sin x
1
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)
f is an odd function

36. The Quotient Rule

37. Surface Area of a Sphere
S = 4 pi r^2
n ln m
Derivative of position at a point
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

38. ln mn
n ln m
ln m - ln n
u'/((ln a) u)
csc²x

39. d/dx[arccot x]
-1/(1+x²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
f(x) g'(x) + g(x) f'(x)
n ln m

40. Area of a circle
pr²
1/(1+x²)
-sin x
v2/2

41. sin p/2
d/dx[cf(x)] = c f'(x)
1
1 / cos x
1/(1+x²)

42. cos 3p/2
0
1 / tan x = cos x / sin x
f is an odd function
sec²x

43. How to get from precalculus to calculus
Limits
u'/((ln a) u)
0
1

44. cos²x + sin²x
0
sec² x
d/dx[x^n]=nx^(n-1)
1

45. sin p/6
sec²x
v3/2
1/2
Derivative of position at a point

46. d/dx[arcsin x]
-csc x cot x
d/dx[cf(x)] = c f'(x)
1/v(1-x²)
1 / tan x = cos x / sin x

47. d/dx[ln x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
V = 4/3 pi r^3
f'(g(x))g'(x)
1/x - x>0

48. sin p/4
v2/2
1/2
(1 + cos 2x) / 2
cos²x - sin²x

49. 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.
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
sec x tan x

50. Position function of a falling object (with acceleration in ft/s²)
d/dx[c] = 0
sec x tan x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
Derivative of position at a point