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 (mn)
ln m + ln n
(1 - cos 2x) / 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


2. Instantaneous velocity
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec²x
(1 - cos 2x) / 2
Derivative of position at a point


3. If f(-x) = f(x)
1 / tan x = cos x / sin 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
f is an even function
Slope of a function at a point/slope of the tangent line to a function at a point


4. Critical number

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5. Volume of a cone
2pr
-1
d/dx[c] = 0
pr²h/3


6. Intermediate Value Theorem
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.
1 / cos x
1/(1+x²)


7. The Quotient Rule

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8. d/dx[arctan x]
1
1/(1+x²)
1 / tan x = cos x / sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.


9. Guidelines for implicit differentiation

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10. d/dx[x]
1
-1
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)


11. d/dx[sec x]
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
Derivative of position at a point
1
sec x tan x


12. tan x
sin x / cos x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
2 sin x cos x
-1/(|x|v(x²-1))


13. d/dx[arcsec x]
0
1 / cos x
csc²x
1/(|x|v(x²-1))


14. If f(-x) = -f(x)
sec²x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
f is an odd function
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


15. sin 0
-1/(|x|v(x²-1))
-1
(ln a) a^x
0


16. d/dx[a^u]

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17. Average speed
u'/((ln a) u)
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.
?s/?t


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

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19. Power Rule for Derivatives
pr²h/3
d/dx[x^n]=nx^(n-1)
0
1/((ln a) x)


20. Indeterminate form
0/0
(ln a) a^x
1
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


21. d/dx[tan x]
f(x) g'(x) + g(x) f'(x)
sec² x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
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)


22. sin p/4
Derivative of position at a point
2pr
v2/2
ln m + ln n


23. ln e
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.
?s/?t
1
u' e^u


24. ln mn
(ln a) a^x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/2
n ln m


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

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26. Area of a circle
1 / tan x = cos x / sin x
0/0
pr²
-csc² x


27. sec x
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
d/dx[cf(x)] = c f'(x)
1 / cos x
sec x tan x


28. sin(2x)
pr²
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
2 sin x cos x
u'/u - u > 0


29. d/dx[arccsc x]
u'/((ln a) u)
cos x
1
-1/(|x|v(x²-1))


30. cot x
pr²
1 / cos x
1 / tan x = cos x / sin x
u'/((ln a) u)


31. Area of an equilateral triangle
1/(1+x²)
e^x
1
v3s² / 4


32. Derivative
e^x
V = 4/3 pi r^3
Slope of a function at a point/slope of the tangent line to a function at a point
1


33. ln 1
-1/v(1-x²)
1/(1+x²)
ln m - ln n
0


34. d/dx[arccos x]
f is an even function
-1/v(1-x²)
pr²h/3
pr²h


35. Derivative of a constant
d/dx[c] = 0
-1
1 / cos x
1/((ln a) x)


36. d/dx[cot x]
-csc² x
1/(|x|v(x²-1))
-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)


37. Position function of a falling object (with acceleration in ft/s²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-csc² x
(1 - cos 2x) / 2
sin x / cos x


38. cos p/6
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)
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'
0


39. cos p
-1
v2/2
?s/?t
u' (ln a) a^u


40. 1 + tan²x
1/2
sec²x
2 sin x cos x
Limits


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

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42. d/dx[sin x]
cos x
csc²x
pr²h
-1


43. d/dx[a^x]
sec x tan x
(ln a) a^x
e^x
v3s² / 4


44. The limit as x approaches 0 of sin x / x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
(1 + cos 2x) / 2
1
f is an odd function


45. Continuity & differentiability
f'(g(x))g'(x)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
?s/?t
(ln a) a^x


46. The limit as x approaches 0 of (1 - cos x) / x
-sin x
0
-1/(|x|v(x²-1))
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


47. Sum and Difference Rules for Derivatives

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48. d/dx[arccot x]
-1/(1+x²)
1
0
1 / cos x


49. d/dx[e^u]

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50. d/dx[e^x]
e^x
Limits
0
u' e^u