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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. d/dx[x]
d/dx[c] = 0
v3/2
(ln a) a^x
1

2. The limit as x approaches 0 of sin x / x
1
e^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
V = 4/3 pi r^3

3. d/dx[e^u]

4. Guidelines for solving related rates problems
1
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
d/dx[c] = 0
0/0

5. The limit as x approaches 0 of (1 - cos x) / x
Derivative of position at a point
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)
u' (ln a) a^u

6. d/dx[ln u]

7. sin p/2
1
sec²x
sin x / cos x
Slope of a function at a point/slope of the tangent line to a function at a point

8. ln 1
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1
u'/u - u > 0
0

9. d/dx[cot x]
-csc² x
1/2
e^x
?s/?t

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

11. Area of a circle
pr²
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
?s/?t
Slope of a function at a point/slope of the tangent line to a function at a point

12. Volume of a right circular cylinder
(1 + cos 2x) / 2
Limits
pr²h
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).

13. sin p
1
sec²x
v3/2
0

14. d/dx[arccsc x]
-1/(|x|v(x²-1))
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
v3/2
-csc² x

15. Intermediate Value Theorem
0
cos²x - sin²x
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'
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.

16. ln (mn)
1
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
ln m + ln n
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)

17. Limit Definition of a Derivative

18. d/dx[log_a u]

19. sin p/4
0
v2/2
1
2pr

20. How to get from precalculus to calculus
Limits
1/2
f is an odd function
Slope of a function at a point/slope of the tangent line to a function at a point

21. Position function of a falling object (with acceleration in ft/s²)
u' e^u
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3s² / 4
0

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

23. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
0
v2/2
1/x - x>0

24. Position function of a falling object (with acceleration in m/s²)
v2/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
Limits
f(x) g'(x) + g(x) f'(x)

25. cos(2x)
Derivative of Position - s'(t)
cos²x - sin²x
1
-csc² x

26. d/dx[arcsec x]
S = 4 pi r^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
d/dx[c] = 0
1/(|x|v(x²-1))

27. 1 + tan²x
1 / sin x
d/dx[c] = 0
u'/((ln a) u)
sec²x

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)
(1 - cos 2x) / 2
V = 4/3 pi r^3
1/2

29. Derivative of a constant
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
d/dx[c] = 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

30. d/dx[a^u]

31. Continuity & differentiability
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
e^x
n ln m

32. Instantaneous velocity
-1/(|x|v(x²-1))
-1
Derivative of position at a point
n ln m

33. sin 0
0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
v2/2
cos²x - sin²x

34. d/dx[a^x]
1
-1/v(1-x²)
(ln a) a^x
Derivative of position at a point

35. Rolle's Theorem

36. d/dx[arccot x]
v3/2
-1/(1+x²)
(1 + cos 2x) / 2
v2/2

37. sin 3p/2
-1
d/dx[x^n]=nx^(n-1)
sec² x
d/dx[cf(x)] = c f'(x)

38. Indeterminate form
1/2
0/0
V = 4/3 pi r^3
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

39. sec x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
ln m - ln n
1 / cos x
n ln m

40. The Quotient Rule

41. d/dx[arccos x]
f is an odd function
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
-1/v(1-x²)
csc²x

42. cot x
1 / tan x = cos x / sin x
0/0
u'/((ln a) u)
u' (ln a) a^u

43. Guidelines for implicit differentiation

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

45. ln e
-1/(1+x²)
csc²x
1
1/(1+x²)

46. cos p/4
0
0
2pr
v2/2

47. d/dx[log_a x]
1/((ln a) x)
1
-csc² x
1

48. d/dx[csc x]
f'(g(x))g'(x)
-csc x cot x
1
v2/2

49. tan x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
Slope of a function at a point/slope of the tangent line to a function at a point
sin x / cos x

50. d/dx[ln 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
1/x - x>0
Derivative of position at a point
1