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. Derivative of a constant
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1 / cos x
d/dx[c] = 0


2. sin 3p/2
-1
pr²h
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
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


3. d/dx[x]
d/dx[cf(x)] = c f'(x)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
f is an even function


4. d/dx[arcsin x]
x values where f'(x) is zero or undefined.
1/v(1-x²)
cos x
1/((ln a) x)


5. 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 - cos 2x) / 2
cos²x - sin²x


6. Indeterminate form
f'(g(x))g'(x)
-1/v(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)
0/0


7. d/dx[cos x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
cos²x - sin²x
-sin x
0


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

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9. The Product Rule

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10. Rolle's Theorem

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11. d/dx[log_a u]

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12. sin p
0
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
(ln a) a^x
1


13. Area of a circle
pr²h/3
pr²
d/dx[x^n]=nx^(n-1)
cos²x - sin²x


14. d/dx[log_a x]
v3/2
-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/((ln a) x)


15. sin p/4
1/v(1-x²)
1 / tan x = cos x / sin x
v2/2
pr²h/3


16. Continuity on a closed interval - [a -b]
cos x
cos²x - sin²x
sin x / cos 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)


17. Power Rule for Derivatives
sec x tan x
0
d/dx[x^n]=nx^(n-1)
-1/(|x|v(x²-1))


18. cos(2x)
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)
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)
cos²x - sin²x


19. ln 1
v3s² / 4
1/(1+x²)
csc²x
0


20. How to get from precalculus to calculus
Limits
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).
ln m - ln n
1 / tan x = cos x / sin x


21. The limit as x approaches 0 of (1 - cos x) / x
-1
1
0
-1/(1+x²)


22. d/dx[arccsc x]
ln m - ln n
d/dx[cf(x)] = c f'(x)
-1/(|x|v(x²-1))
v3/2


23. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
1 / tan x = cos x / sin x
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'
f(x) g'(x) + g(x) f'(x)


24. d/dx[arctan x]
d/dx[c] = 0
1/(1+x²)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


25. Guidelines for solving related rates problems
f(x) g'(x) + g(x) f'(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
f'(g(x))g'(x)
-1/(|x|v(x²-1))


26. d/dx[a^u]

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27. sin²x
sec²x
(1 - cos 2x) / 2
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'(g(x))g'(x)


28. cos p/3
-1/(1+x²)
v2/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/2


29. d/dx[arccos x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
x values where f'(x) is zero or undefined.
-csc² x
-1/v(1-x²)


30. d/dx[ln x]
v3/2
1/x - x>0
d/dx[x^n]=nx^(n-1)
-csc² x


31. d/dx[sin x]
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)
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
cos x


32. The Quotient Rule

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

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34. Mean Value Theorem

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35. d/dx[sec x]
sec x tan x
1/2
(ln a) a^x
-1/(|x|v(x²-1))


36. ln e
2pr
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
ln m - ln n
1


37. d/dx[e^u]

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38. tan x
sin x / cos x
0
f'(g(x))g'(x)
2 sin x cos x


39. cos p/2
1
0
u'/u - u > 0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].


40. sin(2x)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
2 sin x cos x
1
1


41. cot x
cos²x - sin²x
1 / tan x = cos x / sin x
n ln m
csc²x


42. Volume of a Sphere
1/2
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v2/2
V = 4/3 pi r^3


43. d/dx[arccot x]
1
Derivative of position at a point
-1/(1+x²)
d/dx[cf(x)] = c f'(x)


44. If f(-x) = f(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).
1
f is an even function
d/dx[cf(x)] = c f'(x)


45. sin 0
u'/((ln a) u)
?s/?t
S = 4 pi r^2
0


46. cos 0
0
-csc² x
1/2
1


47. d/dx[csc 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)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-csc x cot x
cos²x - sin²x


48. Velocity - v(t)

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49. Guidelines for implicit differentiation

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50. Limit Definition of a Derivative

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