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. ln 1
0
Derivative of position at a point
-1/v(1-x²)
u'/((ln a) u)


2. d/dx[cot x]
1 / sin x
Slope of a function at a point/slope of the tangent line to a function at a point
-csc² x
u' e^u


3. cos p
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
pr²h/3
-1


4. The Quotient Rule

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5. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
-1/v(1-x²)
v3/2
Limits


6. sin p/4
(1 - cos 2x) / 2
-csc x cot x
csc²x
v2/2


7. The limit as x approaches 0 of sin x / x
0
Limits
1
1 / cos x


8. d/dx[arcsin x]
-1
1/v(1-x²)
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)
0


9. Indeterminate form
sec² x
cos²x - sin²x
-1/v(1-x²)
0/0


10. The limit as x approaches 0 of (1 - cos x) / x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
0/0
0
S = 4 pi r^2


11. Guidelines for implicit differentiation

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12. Velocity - v(t)

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13. Volume of a right circular cylinder
cos x
pr²h
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
1


14. d/dx[sec 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
v3/2
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
sec x tan x


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

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16. Continuity at a point (x = c)
v3/2
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.
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)


17. Mean Value Theorem

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

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19. cot x
pr²
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 - cos 2x) / 2
1 / tan x = cos x / sin x


20. Guidelines for solving related rates problems
0
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
1/(|x|v(x²-1))
(1 - cos 2x) / 2


21. cos²x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec²x
(1 + cos 2x) / 2
0


22. If f(-x) = f(x)
f is an even function
1
(1 + cos 2x) / 2
sec x tan x


23. sin²x
1/x - x>0
ln m + ln n
(1 - cos 2x) / 2
2 sin x cos x


24. Position function of a falling object (with acceleration in m/s²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Derivative of position at a point
cos²x - sin²x


25. Alternate Limit Definition of a derivative

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26. d/dx[a^u]

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27. cos(2x)
cos²x - sin²x
1
pr²
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²


28. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v3/2


29. d/dx[arccsc x]
0
1 / cos x
V = 4/3 pi r^3
-1/(|x|v(x²-1))


30. If f(-x) = -f(x)
f'(g(x))g'(x)
v2/2
0
f is an odd function


31. cos p/3
1/2
x values where f'(x) is zero or undefined.
v2/2
0


32. Critical number

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33. sin p
1/((ln a) x)
(1 - cos 2x) / 2
Derivative of position at a point
0


34. Intermediate Value Theorem
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
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.


35. cos²x + sin²x
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)
-csc² x
1/v(1-x²)
1


36. d/dx[sin x]
cos²x - sin²x
cos x
d/dx[x^n]=nx^(n-1)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


37. sin(2x)
v3/2
2 sin x cos x
-sin x
0


38. Circumference of a circle
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
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
2pr


39. sin 0
1
?s/?t
0
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


40. sin p/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)
1
sec² 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.


41. d/dx[arccot x]
-1
sec² x
(ln a) a^x
-1/(1+x²)


42. Power Rule for Derivatives
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/(1+x²)
d/dx[x^n]=nx^(n-1)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


43. Derivative of a constant
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1
d/dx[c] = 0
csc²x


44. Area of an equilateral triangle
v3s² / 4
1 / sin x
csc²x
1/(1+x²)


45. sin 3p/2
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
u'/u - u > 0
u' e^u


46. d/dx[log_a u]

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47. d/dx[a^x]
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
(ln a) a^x
csc²x
-1/v(1-x²)


48. Instantaneous velocity
Derivative of position at a point
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)
-csc x cot x
-1


49. 1 + cot²x
csc²x
sec x tan x
1
-csc x cot x


50. Position function of a falling object (with acceleration in ft/s²)
f(x) g'(x) + g(x) f'(x)
pr²h
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
f'(g(x))g'(x)