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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. 1 + cot²x
f is an even function
-sin x
csc²x
d/dx[cf(x)] = c f'(x)

2. The Quotient Rule

3. ln (mn)
sin x / cos x
sec² x
ln m + ln n
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

4. d/dx[arccos x]
?s/?t
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)
0

5. Limit Definition of a Derivative

6. Mean Value Theorem

7. Guidelines for solving related rates problems
f(x) g'(x) + g(x) f'(x)
1 / tan x = cos x / 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
ln m - ln n

8. 1 + tan²x
f is an even function
S = 4 pi r^2
sec²x
-1

9. 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/(|x|v(x²-1))
0
pr²h/3

10. Critical number

11. d/dx[sin x]
v3/2
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
(ln a) a^x
cos x

12. sin p/3
u'/u - u > 0
1/(|x|v(x²-1))
v3/2
u'/((ln a) u)

13. Volume of a cone
0
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
0
pr²h/3

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

15. cos p/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).
0/0
u' e^u
0

16. Circumference of a circle
2pr
-csc x cot x
u' (ln a) a^u
ln m + ln n

17. sin p
0
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
pr²
d/dx[cf(x)] = c f'(x)

18. How to get from precalculus to calculus
Derivative of position at a point
Limits
0
(1 + cos 2x) / 2

19. Velocity - v(t)

20. d/dx[csc x]
f is an odd function
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
v2/2
-csc x cot x

21. cos 3p/2
(1 + cos 2x) / 2
0
pr²h
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

22. sin 3p/2
-1
sec²x
-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. If f(-x) = -f(x)
f is an odd function
-1/(1+x²)
V = 4/3 pi r^3
1/((ln a) x)

24. d/dx[a^u]

25. cos p/6
1/2
v3/2
1/v(1-x²)
1 / cos x

26. tan x
-1
sec² x
u' (ln a) a^u
sin x / cos x

27. d/dx[tan x]
v3s² / 4
sec² x
u'/((ln a) u)
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'

28. d/dx[a^x]
sec²x
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)
Limits
(ln a) a^x

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

30. d/dx[arcsec x]
ln m - ln n
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/(|x|v(x²-1))
cos x

31. csc 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
2 sin x cos x
1 / sin x
v3/2

32. Average speed
f(x) g'(x) + g(x) f'(x)
1
v3/2
?s/?t

33. d/dx[cos x]
1/2
0
-sin x
2 sin x cos x

34. d/dx[arctan x]
f'(g(x))g'(x)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
0
1/(1+x²)

35. Volume of a Sphere
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
ln m - ln n
V = 4/3 pi r^3

36. cos(2x)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
0
cos²x - sin²x
sec x tan x

37. The limit as x approaches 0 of (1 - cos x) / x
1 / cos x
x values where f'(x) is zero or undefined.
0
pr²h/3

38. The Product Rule

39. d/dx[log_a x]
csc²x
1/((ln a) x)
0
-1

40. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3/2
u' e^u
-1

41. d/dx[e^x]
d/dx[c] = 0
e^x
x values where f'(x) is zero or undefined.
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)

42. sin p/6
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.
1/2
v2/2

43. d/dx[ln x]
1/2
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/x - x>0
Limits

44. Sum and Difference Rules for Derivatives

45. Power Rule for Derivatives
d/dx[x^n]=nx^(n-1)
1/x - x>0
f(x) g'(x) + g(x) f'(x)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

46. ln e
0
f(x) g'(x) + g(x) f'(x)
1
1/x - x>0

47. d/dx[arccot x]
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?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'
Derivative of Position - s'(t)
-1/(1+x²)

48. d/dx[x]
1
d/dx[c] = 0
S = 4 pi r^2
d/dx[cf(x)] = c f'(x)

49. d/dx[cot x]
-csc² x
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
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)

50. sin p/4
1 / tan x = cos x / sin x
f(x) g'(x) + g(x) f'(x)
-1/(1+x²)
v2/2