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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. Power Rule for Derivatives
d/dx[x^n]=nx^(n-1)
pr²h/3
(1 - cos 2x) / 2
0

2. cos p
f'(g(x))g'(x)
-1
0
0/0

3. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
0
(ln a) a^x
1/(1+x²)

4. Mean Value Theorem

5. sin²x
sin x / cos x
d/dx[x^n]=nx^(n-1)
1
(1 - cos 2x) / 2

6. d/dx[arcsec x]
1
1/(|x|v(x²-1))
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)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

7. Circumference of a circle
Derivative of Position - s'(t)
1 / cos x
2pr
0

8. 1 + tan²x
sec²x
-sin x
Derivative of position at a point
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'

9. d/dx[arccos x]
Slope of a function at a point/slope of the tangent line to a function at a point
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1/v(1-x²)
2 sin x cos x

10. ln (m/n)
(1 + cos 2x) / 2
ln m - ln n
0
Limits

11. d/dx[ln u]

12. d/dx[x]
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
sec²x
1

13. Velocity - v(t)

14. Average speed
?s/?t
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
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
0/0

15. Continuity on a closed interval - [a -b]
1/x - x>0
Derivative of Position - s'(t)
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)
d/dx[cf(x)] = c f'(x)

16. d/dx[a^u]

17. Alternate Limit Definition of a derivative

18. The Product Rule

19. Critical number

20. Guidelines for implicit differentiation

21. ln 1
d/dx[cf(x)] = c f'(x)
V = 4/3 pi r^3
2pr
0

22. d/dx[arcsin x]
?s/?t
1 / cos x
1/v(1-x²)
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)

23. d/dx[arccsc x]
-1/(|x|v(x²-1))
f'(g(x))g'(x)
cos x
0

24. Guidelines for solving related rates problems
cos²x - sin²x
1/v(1-x²)
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/(1+x²)

25. sin p/4
v3/2
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v2/2
2pr

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

27. cos²x
v3s² / 4
(1 + cos 2x) / 2
ln m - ln n
(1 - cos 2x) / 2

28. Derivative of a constant
pr²
d/dx[c] = 0
-1/v(1-x²)
1

29. sec x
1 / cos x
1/((ln a) x)
1/x - x>0
0

30. sin(2x)
pr²h/3
2 sin x cos x
1
-1

31. Surface Area of a Sphere
1
1/(1+x²)
S = 4 pi r^2
-1/(|x|v(x²-1))

32. Extreme Value Theorem
0
f is an even function
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
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

33. Position function of a falling object (with acceleration in ft/s²)
-1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
(ln a) a^x
1 / tan x = cos x / sin x

34. sin 0
1/2
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).
0
u' (ln a) a^u

35. If f(-x) = -f(x)
f is an odd function
sin x / cos x
f'(g(x))g'(x)
(1 - cos 2x) / 2

36. Continuity at a point (x = c)
f is an odd function
-sin 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. 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)

37. Indeterminate form
-1
Derivative of Position - s'(t)
sec² x
0/0

38. Area of an equilateral triangle
v3/2
v3s² / 4
0
n ln m

39. Derivative of an inverse (if g(x) is the inverse of f(x))

40. The Quotient Rule

41. cos(2x)
1 / tan x = cos x / sin x
cos²x - sin²x
ln m - ln n
-sin x

42. Limit Definition of a Derivative

43. sin p/3
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
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3/2
-1/(1+x²)

44. d/dx[sec x]
x values where f'(x) is zero or undefined.
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
sec x tan x
(1 + cos 2x) / 2

45. cos p/4
1/v(1-x²)
Derivative of position at a point
v2/2
sec² x

46. tan x
sin x / cos x
1 / tan x = cos x / sin 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'
Limits

47. d/dx[a^x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
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
1/2
(ln a) a^x

48. sin p/6
sec² x
pr²h
1/2
-csc² x

49. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
0/0
1
sec²x

50. cot x
1 / tan x = cos x / sin x
0
1/x - x>0
v3/2