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[cos x]
Derivative of position at a point
d/dx[x^n]=nx^(n-1)
0
-sin x


2. d/dx[x]
Limits
1
-csc x cot x
1/2


3. ln 1
1/2
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'
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
0


4. cos(2x)
e^x
-1/(|x|v(x²-1))
cos²x - sin²x
1 / tan x = cos x / sin x


5. csc x
sin x / cos x
0
1/2
1 / sin x


6. cos 0
1
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
pr²h/3
f'(g(x))g'(x)


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


8. Continuity on a closed interval - [a -b]
V = 4/3 pi r^3
-csc x cot 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)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x


9. cos 3p/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)
x values where f'(x) is zero or undefined.
n ln m
0


10. Rolle's Theorem

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11. d/dx[arcsec x]
-csc² x
pr²h/3
1/(|x|v(x²-1))
f is an odd function


12. 1 + tan²x
u' (ln a) a^u
sec²x
0
1


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

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14. Position function of a falling object (with acceleration in ft/s²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/v(1-x²)
1
cos x


15. d/dx[ln x]
1/x - x>0
v2/2
1/(|x|v(x²-1))
1/(1+x²)


16. sin p/2
0
1
0
0


17. Alternate Limit Definition of a derivative

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18. Indeterminate form
0/0
1/2
(ln a) a^x
u'/u - u > 0


19. Volume of a cone
1
u'/((ln a) u)
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
pr²h/3


20. sin p/4
v2/2
f'(g(x))g'(x)
0/0
-1


21. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
d/dx[x^n]=nx^(n-1)
u'/((ln a) u)
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.


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

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23. Derivative of a constant
1 / tan x = cos x / sin x
f is an even function
d/dx[c] = 0
-sin x


24. cot x
1 / tan x = cos x / sin x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
-sin x
1


25. Surface Area of a Sphere
cos²x - sin²x
1
S = 4 pi r^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)


26. d/dx[arccsc x]
-1
0/0
2pr
-1/(|x|v(x²-1))


27. cos p/4
0
e^x
v2/2
-csc² x


28. The limit as x approaches 0 of (1 - cos x) / x
v3s² / 4
0
S = 4 pi r^2
1


29. Area of an equilateral triangle
2pr
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
2 sin x cos x
v3s² / 4


30. sec x
1 / cos x
-csc² x
0
2 sin x cos x


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

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32. d/dx[sin x]
-1/(1+x²)
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'
-1
cos x


33. sin²x
d/dx[cf(x)] = c f'(x)
u'/u - u > 0
(1 - cos 2x) / 2
ln m - ln n


34. sin 0
Derivative of position at a point
csc²x
1
0


35. Continuity on an open interval - (a -b)
-1/(|x|v(x²-1))
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
ln m + ln n
-csc x cot x


36. Instantaneous velocity
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
Limits
Derivative of position at a point
-1


37. sin p/6
1/2
1
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'
1


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


39. ln mn
e^x
sec²x
v3/2
n ln m


40. d/dx[e^x]
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-csc x cot x
e^x
d/dx[x^n]=nx^(n-1)


41. d/dx[tan x]
sec² x
1 / sin x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
u'/((ln a) u)


42. cos²x
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
f is an odd function
2pr


43. d/dx[e^u]

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44. d/dx[sec x]
1 / tan x = cos x / sin x
1
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
sec x tan x


45. ln e
1/((ln a) x)
1
d/dx[x^n]=nx^(n-1)
Derivative of position at a point


46. Power Rule for Derivatives
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.
pr²
d/dx[x^n]=nx^(n-1)
sec²x


47. cos p/2
0
1/2
u' e^u
cos²x - sin²x


48. The Quotient Rule

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49. Constant Multiple Rule for Derivatives

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50. d/dx[log_a x]
-1/v(1-x²)
1/((ln a) 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).
V = 4/3 pi r^3