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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. 1 + cotx
prh/3
pr
cscx
?s/?t

2. If f(-x) = f(x)
1
v2/2
f is an even function
Derivative of Position - s'(t)

3. Volume of a Sphere
cos x
V = 4/3 pi r^3
1/2
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height

4. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
1 / cos x
1/((ln a) x)
-csc x

5. sin p/6
1/2
0
0
cscx

6. cos p/6
f'(g(x))g'(x)
1
v3/2
f(x) g'(x) + g(x) f'(x)

7. Alternate Limit Definition of a derivative
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
secx
0

8. d/dx[arcsec 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)
cos x
1/v(1-x)
1/(|x|v(x-1))

9. d/dx[ln x]
-csc x
1/x - x>0
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

10. Surface Area of a Sphere
Limits
f(x) g'(x) + g(x) f'(x)
u'/u - u > 0
S = 4 pi r^2

11. Position function of a falling object (with acceleration in m/s)
v3/2
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
u'/((ln a) u)

12. Velocity - v(t)
u'/((ln a) u)
0
1
Derivative of Position - s'(t)

13. d/dx[e^x]
d/dx[x^n]=nx^(n-1)
pr
v2/2
e^x

14. sin p/3
ln m + ln n
1
1 / tan x = cos x / sin x
v3/2

15. ln e
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)]
sec x
1
pr

16. d/dx[ f(x) / g(x) ]
1 / sin x
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

17. d/dx[a^u]
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
sec x tan x
u' (ln a) a^u

18. Circumference of a circle
sec x
Slope of a function at a point/slope of the tangent line to a function at a point
0/0
2pr

19. sin 3p/2
ln m - ln n
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height
0
-1

20. d/dx[tan x]
f'(g(x))g'(x)
-1
e^x
sec x

21. csc x
1 / sin x
f'(g(x))g'(x)
0
1

22. cos p/2
s(t) = -16t+ v0t + s0 - v0 = initial velocity - s0 = initial height
(1 - cos 2x) / 2
cosx - sinx
0

23. d/dx[arcsin x]
1/v(1-x)
e^x
0
cosx - sinx

24. sec x
secx
1 / cos x
d/dx[f(x) g(x)] = f'(x) g'(x)
1/(|x|v(x-1))

25. d/dx[log_a x]
f'(g(x))g'(x)
1 / sin x
1/((ln a) x)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

26. Intermediate Value Theorem
v3/2
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

27. Indeterminate form
0/0
0
Derivative of position at a point
S = 4 pi r^2

28. Continuity on a closed interval - [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)
ln m + ln n
1 / cos x
sec x

29. Critical number
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
x values where f'(x) is zero or undefined.
-1/(|x|v(x-1))
-1/(1+x)

30. sin(2x)
2 sin x cos 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)
0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

31. Area of an equilateral triangle
v3s / 4
1/x - x>0
0
1

32. Derivative of a constant
sec x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
cos x
d/dx[c] = 0

33. cos p/3
1/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1 / cos x
secx

34. cot x
1 / tan x = cos x / sin x
Limits
1 / sin x
u' (ln a) a^u

35. cos(2x)
f is an odd function
d/dx[f(x) g(x)] = f'(x) g'(x)
cosx - sinx
1/x - x>0

36. The Product Rule
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
1/((ln a) x)
-1/v(1-x)
1 / cos x

37. Power Rule for Derivatives
1 / tan x = cos x / sin x
ln m + ln n
d/dx[x^n]=nx^(n-1)
n ln m

38. If f(-x) = -f(x)
Derivative of position at a point
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
f is an even function
f is an odd function

39. Position function of a falling object (with acceleration in ft/s)
s(t) = -16t+ 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)
x values where f'(x) is zero or undefined.
?s/?t

40. d/dx[ln u]
ln m + ln n
cos x
u'/u - u > 0
d/dx[c] = 0

41. Mean Value Theorem
?s/?t
pr
0
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)

42. Continuity at a point (x = c)
f'(g(x))g'(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)
2pr
2 sin x cos x

43. d/dx[ f(x) g(x) ]
f(x) g'(x) + g(x) f'(x)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 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).

44. Guidelines for implicit differentiation
0
2pr
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'

45. sin 0
d/dx[f(x) g(x)] = f'(x) g'(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)
0
prh/3

46. cos p/4
-1/(|x|v(x-1))
v2/2
x values where f'(x) is zero or undefined.
d/dx[cf(x)] = c f'(x)

47. d/dx[csc x]
0
-csc x cot 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)
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).

48. cos p
f(x) g'(x) + g(x) f'(x)
(1 - cos 2x) / 2
-1
cosx - sinx

49. d/dx[cos x]
sec x tan x
-sin x
prh
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height

50. sinx
0
v2/2
sec x
(1 - cos 2x) / 2

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