AP Calculus Formulas

Instructions:

• Answer 50 questions in 15 minutes.
• If you are not ready to take this test, you can study here.
• Match each statement with the correct term.
• Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. sin p/6
sec² x
1/2
1/(|x|v(x²-1))
1 / tan x = cos x / sin x


2. Area of a circle
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)
pr²
ln m - ln n


3. Position function of a falling object (with acceleration in ft/s²)
csc²x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
Limits


4. d/dx[e^u]

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5. ln mn
n ln m
1
-1/(|x|v(x²-1))
d/dx[c] = 0


6. d/dx[arccot x]
-1/(1+x²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
Limits
x values where f'(x) is zero or undefined.


7. The limit as x approaches 0 of sin x / x
u' e^u
1 / cos x
V = 4/3 pi r^3
1


8. Power Rule for Derivatives
1 / sin x
d/dx[x^n]=nx^(n-1)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
1/2


9. sin p
sec x tan x
-1/v(1-x²)
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²


10. 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.
sin x / cos x
-1
-csc x cot x


11. cos 3p/2
sin x / cos x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1
0


12. d/dx[sin x]
n ln m
cos x
-sin x
0


13. d/dx[log_a x]
1/((ln a) x)
sec²x
1/x - x>0
f is an odd function


14. sin p/3
sec x tan x
d/dx[c] = 0
v3/2
Derivative of Position - s'(t)


15. tan x
sin x / cos x
(ln a) a^x
0
-1


16. Volume of a Sphere
0
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.
v3s² / 4
V = 4/3 pi r^3


17. Volume of a cone
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. 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)
pr²h/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


18. Alternate Limit Definition of a derivative

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19. Volume of a right circular cylinder
-csc x cot x
x values where f'(x) is zero or undefined.
2 sin x cos x
pr²h


20. ln (m/n)
u' e^u
-1
v3/2
ln m - ln n


21. Indeterminate form
v2/2
0/0
sec² x
V = 4/3 pi r^3


22. Velocity - v(t)

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23. ln 1
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
0
d/dx[cf(x)] = c f'(x)


24. Derivative
Limits
f is an odd function
1
Slope of a function at a point/slope of the tangent line to a function at a point


25. How to get from precalculus to calculus
2 sin x cos x
Limits
1/x - x>0
e^x


26. sin 3p/2
d/dx[c] = 0
-1/(1+x²)
-1
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'


27. sin p/2
1
0
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.
d/dx[cf(x)] = c f'(x)


28. Circumference of a circle
1
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)
2pr
Limits


29. Sum and Difference Rules for Derivatives

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30. 1 + tan²x
(1 - cos 2x) / 2
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
sec²x
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)


31. sec x
e^x
1/x - x>0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1 / cos x


32. d/dx[sec x]
sec x tan x
0
d/dx[cf(x)] = c f'(x)
-1/v(1-x²)


33. cos²x
v2/2
pr²h/3
ln m + ln n
(1 + cos 2x) / 2


34. cos(2x)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
pr²h
v2/2
cos²x - sin²x


35. cos p/4
f is an odd function
v2/2
u'/((ln a) u)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


36. Guidelines for solving related rates problems
0
1
csc²x
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


37. ln e
ln m + ln n
1
-1
n ln m


38. Critical number

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39. Average speed
2 sin x cos x
(1 - cos 2x) / 2
?s/?t
u' e^u


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

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41. d/dx[arcsec x]
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
(1 + cos 2x) / 2
u'/((ln a) u)
1/(|x|v(x²-1))


42. Instantaneous velocity
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1
Derivative of position at a point
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²


43. Extreme Value Theorem
-1/v(1-x²)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
-1
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²


44. Rolle's Theorem

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45. d/dx[cot x]
2pr
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
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)
-csc² x


46. 1 + cot²x
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
?s/?t
csc²x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


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

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48. Mean Value Theorem

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49. sin 0
-1/(|x|v(x²-1))
0
1/(1+x²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


50. d/dx[arcsin x]
u'/u - u > 0
1/v(1-x²)
0
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)