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. Instantaneous velocity
v3s² / 4
(ln a) a^x
1/((ln a) x)
Derivative of position at a point


2. Velocity - v(t)

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3. Rolle's Theorem

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4. The Product Rule

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5. cos 0
1/(|x|v(x²-1))
v3/2
1
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)


6. d/dx[arccsc x]
S = 4 pi r^2
0
u'/u - u > 0
-1/(|x|v(x²-1))


7. csc x
0
(1 + cos 2x) / 2
pr²h/3
1 / sin x


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

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9. d/dx[a^x]
(ln a) a^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
1
v3/2


10. sin p/6
f(x) g'(x) + g(x) f'(x)
0
-sin x
1/2


11. cos p/2
-sin x
0
u'/((ln a) u)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x


12. sin p
1/((ln a) x)
0
e^x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


13. d/dx[csc x]
-csc x cot x
1 / cos x
-1
v3/2


14. Guidelines for solving related rates problems
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
u' e^u
n ln m
f(x) g'(x) + g(x) f'(x)


15. Volume of a Sphere
V = 4/3 pi r^3
pr²h/3
-sin x
d/dx[c] = 0


16. Derivative
-1
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)
Slope of a function at a point/slope of the tangent line to a function at a point
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)


17. d/dx[ln x]
1/x - x>0
f is an even function
v2/2
Derivative of Position - s'(t)


18. The Quotient Rule

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19. The limit as x approaches 0 of sin x / x
1
1/((ln a) x)
pr²
v3/2


20. Sum and Difference Rules for Derivatives

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21. sin p/2
1/x - x>0
1
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
0


22. cos p/4
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u'/u - u > 0
Limits
v2/2


23. Position function of a falling object (with acceleration in ft/s²)
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)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
(1 + cos 2x) / 2
f(x) g'(x) + g(x) f'(x)


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


25. Intermediate Value Theorem
1/2
V = 4/3 pi r^3
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.
pr²h/3


26. d/dx[arcsec x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
1
1 / tan x = cos x / sin x
1/(|x|v(x²-1))


27. sec x
1 / cos 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).
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


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

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29. cot x
(1 - cos 2x) / 2
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
-sin x
1 / tan x = cos x / sin x


30. d/dx[ln u]

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31. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m + ln n
V = 4/3 pi r^3
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


32. ln (mn)
f is an even function
0
2pr
ln m + ln n


33. How to get from precalculus to calculus
sec²x
f is an odd function
Limits
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


34. cos p
Derivative of position at a point
pr²h
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
-1


35. d/dx[e^u]

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36. Guidelines for implicit differentiation

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37. d/dx[e^x]
e^x
v2/2
f(x) g'(x) + g(x) f'(x)
csc²x


38. Indeterminate form
0/0
v2/2
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)
1/2


39. Mean Value Theorem

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40. Extreme Value Theorem
1 / tan x = cos x / sin x
1 / cos x
sec²x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].


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

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42. d/dx[x]
1 / cos x
1/((ln a) x)
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.


43. ln e
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)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1 / tan x = cos x / sin x


44. d/dx[arccos x]
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.
ln m - ln n
-1/v(1-x²)
sin x / cos x


45. d/dx[sec x]
1
sec x tan x
-1/v(1-x²)
u'/u - u > 0


46. Continuity & differentiability
0
-1/(|x|v(x²-1))
v2/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.


47. sin 3p/2
-sin x
?s/?t
u' (ln a) a^u
-1


48. Surface Area of a Sphere
-1
S = 4 pi r^2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
Derivative of position at a point


49. d/dx[log_a u]

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50. tan x
sin x / cos x
1
sec x tan x
v2/2