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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. cos 3p/2
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
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'

2. sin p
0
?s/?t
1
1 / tan x = cos x / sin x

3. Mean Value Theorem

4. d/dx[arcsec x]
0
1/((ln a) x)
v2/2
1/(|x|v(x²-1))

5. d/dx[a^x]
Derivative of Position - s'(t)
(ln a) a^x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
-csc² x

6. Surface Area of a Sphere
1
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
S = 4 pi r^2
f'(g(x))g'(x)

7. sin p/6
1/(1+x²)
f is an odd function
1/2
1/v(1-x²)

8. d/dx[sin x]
cos²x - sin²x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-1
cos x

9. Guidelines for implicit differentiation

10. sin p/4
-csc x cot x
v2/2
?s/?t
Slope of a function at a point/slope of the tangent line to a function at a point

11. Continuity on a closed interval - [a -b]
f'(g(x))g'(x)
1
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
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)

12. 1 + cot²x
csc²x
(ln a) a^x
u' e^u
f'(g(x))g'(x)

13. Critical number

14. Guidelines for solving related rates problems
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
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

15. d/dx[cot x]
1
-1/v(1-x²)
-csc² x
(ln a) a^x

16. ln (m/n)
ln m - ln n
1
2 sin x cos x
1/2

17. d/dx[sec x]
e^x
1/(1+x²)
sec x tan x
sin x / cos x

18. cos p/2
0
-1/v(1-x²)
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 - cos 2x) / 2

19. The Product Rule

20. Area of an equilateral triangle
-1
v3s² / 4
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
2 sin x cos x

21. ln e
1
sec x tan x
cos 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)

22. sin(2x)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
2 sin x cos x
d/dx[cf(x)] = c f'(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)

23. cos p/6
v3/2
d/dx[x^n]=nx^(n-1)
sec² x
1

24. Volume of a Sphere
V = 4/3 pi r^3
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
d/dx[cf(x)] = c f'(x)
u' (ln a) a^u

25. d/dx[arctan x]
v2/2
1/(1+x²)
(1 - cos 2x) / 2
S = 4 pi r^2

26. Average speed
sin x / cos x
sec x tan 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).
?s/?t

27. cos 0
1
?s/?t
pr²
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

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

29. d/dx[log_a u]

30. If f(-x) = f(x)
1 / cos x
v3/2
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f is an even function

31. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
f'(g(x))g'(x)
1

32. Circumference of a circle
2pr
1 / tan x = cos x / sin x
0
f(x) g'(x) + g(x) f'(x)

33. Indeterminate form
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
?s/?t
0/0
1

34. sec x
1
1 / cos x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
sec² x

35. Area of a circle
-csc x cot x
1
pr²
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

36. d/dx[cos x]
-1/(|x|v(x²-1))
-sin x
(1 - cos 2x) / 2
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)

37. ln (mn)
d/dx[x^n]=nx^(n-1)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
ln m + ln n
1/(1+x²)

38. 1 + tan²x
u' e^u
sec²x
1 / tan x = cos x / sin x
Derivative of Position - s'(t)

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

40. d/dx[arccos x]
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-1/v(1-x²)
n ln m
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)

41. Derivative of a constant
d/dx[c] = 0
1 / sin x
cos²x - sin²x
-1/(|x|v(x²-1))

42. d/dx[csc 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)
f is an even function
-csc x cot x
f is an odd function

43. d/dx[tan x]
sec² x
pr²h/3
n ln m
Slope of a function at a point/slope of the tangent line to a function at a point

44. Derivative
1 / sin x
sec² x
sec²x
Slope of a function at a point/slope of the tangent line to a function at a point

45. Position function of a falling object (with acceleration in ft/s²)
1/v(1-x²)
1 / tan x = cos x / sin x
x values where f'(x) is zero or undefined.
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

46. The limit as x approaches 0 of sin x / x
cos²x - sin²x
e^x
d/dx[x^n]=nx^(n-1)
1

47. The limit as x approaches 0 of (1 - cos x) / x
0
csc²x
pr²
1 / sin x

48. cos p
f'(g(x))g'(x)
-1
-1/(1+x²)
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)

49. Position function of a falling object (with acceleration in m/s²)
0
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
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.
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. d/dx[x]
1
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).
cos x
Slope of a function at a point/slope of the tangent line to a function at a point