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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. cos 3p/2
(ln a) a^x
0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1/x - x>0

2. Power Rule for Derivatives
d/dx[x^n]=nx^(n-1)
prh/3
f is an even function
2pr

3. Indeterminate form
1 / cos x
0/0
0
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

4. If f(-x) = f(x)
u'/((ln a) u)
2 sin x cos x
1/(1+x)
f is an even function

5. The Quotient Rule
1 / cos x
?s/?t
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)]
prh/3

6. ln 1
-csc x cot x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
1/(1+x)
0

7. sec x
1 / cos 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
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

8. ln e
?s/?t
ln m + ln n
1
x values where f'(x) is zero or undefined.

9. The Product Rule
ln m - ln n
-1/(|x|v(x-1))
0
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

10. Area of an equilateral triangle
2pr
v3/2
v3s / 4
2 sin x cos x

11. How to get from precalculus to calculus
cosx - sinx
-csc x
Limits
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.

12. d/dx[arccos x]
-1/v(1-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).
-1
sin x / cos x

13. ln (mn)
ln m + ln n
s(t) = -16t+ v0t + s0 - v0 = initial velocity - s0 = initial height
0
u' (ln a) a^u

14. cos p
v2/2
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)
-1
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height

15. d/dx[log_a x]
1/((ln a) x)
sin x / cos x
-sin x
ln m + ln n

16. Derivative of an inverse (if g(x) is the inverse of f(x))
u' (ln a) a^u
?s/?t
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
0

17. d/dx[sec x]
0
n ln m
sec x tan x
u' (ln a) a^u

18. cos(2x)
Derivative of Position - s'(t)
cosx - sinx
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/x - x>0

19. sin p/3
secx
v3/2
0
prh/3

20. d/dx[cot x]
1/((ln a) x)
2pr
1
-csc x

21. ln mn
0
d/dx[cf(x)] = c f'(x)
u' e^u
n ln m

22. Instantaneous velocity
ln m + ln n
d/dx[c] = 0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
Derivative of position at a point

23. d/dx[csc x]
1
0
-csc x cot 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)

24. d/dx[e^x]
e^x
u' e^u
u' (ln a) a^u
ln m - ln n

25. Volume of a right circular cylinder
prh
1
f'(g(x))g'(x)
d/dx[x^n]=nx^(n-1)

26. d/dx[tan x]
V = 4/3 pi r^3
1
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)
sec x

27. d/dx[a^u]
u' (ln a) a^u
cos x
1
v3s / 4

28. sinx
1
(1 - cos 2x) / 2
v2/2
x values where f'(x) is zero or undefined.

29. ln (m/n)
ln m - ln n
1/v(1-x)
1
-csc x cot x

30. d/dx[arcsec x]
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
2pr
1/(|x|v(x-1))
1 / cos x

31. cosx + sinx
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/((ln a) x)
-1/(1+x)

32. d/dx[arcsin x]
1/(|x|v(x-1))
-1/(|x|v(x-1))
-1/(1+x)
1/v(1-x)

33. sin p/6
1/2
2 sin x cos x
sin x / cos x
v2/2

34. 1 + tanx
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
secx
0
f(x) g'(x) + g(x) f'(x)

35. cos 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
0
1
0

36. d/dx[e^u]
u' e^u
v3s / 4
-1/(|x|v(x-1))
ln m + ln n

37. d/dx[ln u]
-csc x
u'/u - u > 0
v3/2
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).

38. Average speed
f is an odd function
?s/?t
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
cscx

39. Velocity - v(t)
1 / tan x = cos x / sin x
1
Derivative of Position - s'(t)
1/2

40. Continuity & differentiability
pr
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
sin x / cos x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

41. Alternate Limit Definition of a derivative
-1/(|x|v(x-1))
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
sin x / cos x

42. d/dx[arctan x]
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
1/(1+x)
1 / tan x = cos x / sin x
prh/3

43. tan x
sin x / cos x
d/dx[cf(x)] = c f'(x)
1 / cos x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

44. Critical number
1 / tan x = cos x / sin x
-1
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
x values where f'(x) is zero or undefined.

45. sin p/4
-csc x
1/x - x>0
1
v2/2

46. Volume of a Sphere
n ln m
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
V = 4/3 pi r^3
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height

47. The limit as x approaches 0 of (1 - cos x) / x
sec x
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
0
1/(1+x)

48. Position function of a falling object (with acceleration in m/s)
u'/u - u > 0
1 / sin 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)
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height

49. The limit as x approaches 0 of sin x / x
0
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
-csc x
1

50. Guidelines for implicit differentiation
(1 + cos 2x) / 2
d/dx[cf(x)] = c f'(x)
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'
0

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