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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. cos 3p/2
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)
(ln a) a^x
0
2 sin x cos x

2. Rolle's Theorem

3. Velocity - v(t)

4. d/dx[a^x]
(ln a) a^x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
2 sin x cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

5. Position function of a falling object (with acceleration in ft/s²)
1/v(1-x²)
x values where f'(x) is zero or undefined.
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
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)

6. cos²x + sin²x
1/2
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f(x) g'(x) + g(x) f'(x)
1

7. d/dx[tan x]
1/(|x|v(x²-1))
?s/?t
sec² 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).

8. sin p
-1/(|x|v(x²-1))
S = 4 pi r^2
0
-sin x

9. cos p
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
-1
-csc² x

10. Continuity at a point (x = c)
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 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. 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

11. sin²x
v3/2
(1 - cos 2x) / 2
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
u'/u - u > 0

12. d/dx[a^u]

13. The limit as x approaches 0 of (1 - cos x) / x
u'/u - u > 0
0
2 sin x cos x
1

14. sin p/6
-csc x cot x
f(x) g'(x) + g(x) f'(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

15. d/dx[ln u]

16. Volume of a Sphere
1/2
V = 4/3 pi r^3
2pr
1/v(1-x²)

17. d/dx[x]
sec² x
2pr
1
u' e^u

18. Guidelines for solving related rates problems
pr²
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
e^x
1/((ln a) x)

19. The limit as x approaches 0 of sin x / x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
d/dx[x^n]=nx^(n-1)
1/v(1-x²)

20. cos p/4
v2/2
u' (ln a) a^u
1
(1 - cos 2x) / 2

21. d/dx[arccsc x]
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1/(|x|v(x²-1))
1/x - x>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)

22. sin 0
-1
1
u' e^u
0

23. sin p/4
(ln a) a^x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
Limits
v2/2

24. If f(-x) = f(x)
n ln m
(1 + cos 2x) / 2
0
f is an even function

25. Limit Definition of a Derivative

26. 1 + cot²x
sec² x
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)
csc²x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

27. d/dx[e^x]
v2/2
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)
e^x
sec x tan x

28. tan x
pr²
-csc x cot x
1/v(1-x²)
sin x / cos x

29. sin 3p/2
v2/2
1/2
-1
S = 4 pi r^2

30. d/dx[ln x]
u'/((ln a) u)
-1
1
1/x - x>0

31. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
ln m + ln n
1
Limits

32. How to get from precalculus to calculus
Limits
n ln m
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

33. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
d/dx[c] = 0
1/(1+x²)
f is an even function

34. Area of a circle
1
-1
pr²
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

35. d/dx[arcsec x]
d/dx[c] = 0
f'(g(x))g'(x)
-sin x
1/(|x|v(x²-1))

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

37. d/dx[sin 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
pr²h/3
Limits
cos x

38. Surface Area of a Sphere
S = 4 pi r^2
d/dx[c] = 0
pr²h
-1

39. Sum and Difference Rules for Derivatives

40. Continuity on an open interval - (a -b)
?s/?t
2 sin x cos x
1 / tan x = cos x / sin x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

41. d/dx[arccos x]
pr²
-1/v(1-x²)
pr²h/3
Derivative of Position - s'(t)

42. d/dx[csc x]
-csc x cot x
pr²h
1
0

43. d/dx[log_a x]
1
sec²x
1/((ln a) x)
u' e^u

44. d/dx[cot x]
-csc² x
csc²x
v3/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

45. d/dx[arcsin x]
cos²x - sin²x
0/0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1/v(1-x²)

46. cot x
1 / tan x = cos x / sin x
Derivative of position at a point
0
v2/2

47. Derivative of a constant
d/dx[c] = 0
-1
f is an even function
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

48. Mean Value Theorem

49. Average speed
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
sin x / cos x
?s/?t
0/0

50. 1 + tan²x
-1
d/dx[cf(x)] = c f'(x)
sec²x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]