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. d/dx[csc x]
S = 4 pi r^2
1
-csc² x
-csc x cot x


2. 1 + tan²x
0/0
1
sec²x
sec² x


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


4. Sum and Difference Rules for Derivatives

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5. Area of an equilateral triangle
v3s² / 4
0
v3/2
cos x


6. Critical number

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7. Mean Value Theorem

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8. Volume of a Sphere
f is an odd function
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.
V = 4/3 pi r^3
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. cos²x + sin²x
1/(|x|v(x²-1))
f is an even function
u'/u - u > 0
1


10. d/dx[arccsc x]
-1/(|x|v(x²-1))
pr²h
0
1 / sin x


11. cos 0
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 / tan x = cos x / sin x
ln m + ln n


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

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

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14. Rolle's Theorem

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15. d/dx[e^u]

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16. Circumference of a circle
-1/(|x|v(x²-1))
2pr
1
V = 4/3 pi r^3


17. Derivative
f'(g(x))g'(x)
Slope of a function at a point/slope of the tangent line to a function at a point
1/v(1-x²)
-sin x


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

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19. d/dx[sin x]
Limits
cos x
f is an even function
(1 + cos 2x) / 2


20. Volume of a right circular cylinder
v2/2
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).
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)


21. Continuity & differentiability
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


22. sin 0
-sin x
0
-1/(1+x²)
2 sin x cos x


23. Power Rule for Derivatives
d/dx[x^n]=nx^(n-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)
v3/2
ln m - ln n


24. Derivative of a constant
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 / sin x
pr²h
d/dx[c] = 0


25. d/dx[arcsec x]
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
1
1/(|x|v(x²-1))
sec²x


26. cos p/6
f is an even function
-1/(|x|v(x²-1))
v3/2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


27. Average speed
d/dx[x^n]=nx^(n-1)
ln m + ln n
?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.


28. d/dx[arccos x]
0
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
csc²x
-1/v(1-x²)


29. sin p/4
v2/2
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
-csc² x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


30. d/dx[x]
0
1
f is an odd function
pr²h/3


31. d/dx[cot x]
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
-csc² x
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
Limits


32. If f(-x) = f(x)
1
(1 - cos 2x) / 2
1/((ln a) x)
f is an even function


33. d/dx[arccot x]
cos x
1 / tan x = cos x / sin x
0
-1/(1+x²)


34. d/dx[a^x]
(ln a) a^x
-csc² x
n ln m
1/2


35. Guidelines for implicit differentiation

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36. d/dx[sec x]
1/x - x>0
x values where f'(x) is zero or undefined.
sec x tan x
f(x) g'(x) + g(x) f'(x)


37. cos(2x)
f is an even function
x values where f'(x) is zero or undefined.
cos²x - sin²x
Slope of a function at a point/slope of the tangent line to a function at a point


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


39. sin(2x)
2 sin x cos x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
Slope of a function at a point/slope of the tangent line to a function at a point
pr²h


40. Intermediate Value Theorem
-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.
f is an even function
1/2


41. d/dx[e^x]
1 / tan x = cos x / sin x
u'/u - u > 0
(ln a) a^x
e^x


42. sin²x
ln m + ln n
(1 - cos 2x) / 2
sec x tan x
1/(1+x²)


43. sin p/3
pr²h/3
-csc² x
v3/2
ln m + ln n


44. Volume of a cone
pr²h/3
ln m - ln n
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
-csc² x


45. d/dx[log_a x]
cos²x - sin²x
1/((ln a) x)
-csc x cot x
S = 4 pi r^2


46. d/dx[a^u]

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47. Velocity - v(t)

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48. d/dx[arcsin x]
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
S = 4 pi r^2
1/v(1-x²)


49. cos p/4
v2/2
e^x
v3/2
sec x tan x


50. 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)
u' (ln a) a^u
2 sin x cos x
0