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. sin p/6
1/x - x>0
1/2
(1 - cos 2x) / 2
-csc² x


2. sec x
f(x) g'(x) + g(x) f'(x)
u'/((ln a) u)
f is an odd function
1 / cos x


3. 1 + cot²x
Derivative of Position - s'(t)
csc²x
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)


4. Mean Value Theorem

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5. Continuity on a closed interval - [a -b]
sec x tan x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
0
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)


6. The limit as x approaches 0 of sin x / x
d/dx[cf(x)] = c f'(x)
v3s² / 4
1/(|x|v(x²-1))
1


7. Instantaneous velocity
1 / sin x
sec²x
Derivative of position at a point
0


8. d/dx[tan x]
sec² x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
0
d/dx[x^n]=nx^(n-1)


9. Derivative
ln m - ln n
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.
Slope of a function at a point/slope of the tangent line to a function at a point
1/x - x>0


10. d/dx[sin x]
cos x
1
pr²
-1/(1+x²)


11. If f(-x) = f(x)
v3s² / 4
pr²
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 is an even function


12. Sum and Difference Rules for Derivatives

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13. sin 3p/2
f'(g(x))g'(x)
-csc² x
1
-1


14. cos p/6
v3/2
-1/(1+x²)
-1
sin x / cos x


15. sin²x
-sin x
(1 - cos 2x) / 2
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
v3/2


16. d/dx[log_a x]
f is an odd function
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/((ln a) x)
u' e^u


17. Rolle's Theorem

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18. d/dx[arccot x]
2 sin x cos x
f(x) g'(x) + g(x) f'(x)
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/(1+x²)


19. Continuity on an open interval - (a -b)
0
v3s² / 4
2 sin x cos x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


20. d/dx[arcsec x]
1/(|x|v(x²-1))
v2/2
f is an even function
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'


21. sin p
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
0
?s/?t
ln m + ln n


22. d/dx[cos x]
(1 - cos 2x) / 2
-sin 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).
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


23. cos²x + sin²x
0
1
-1/(|x|v(x²-1))
v3s² / 4


24. cos p/3
1
(1 + cos 2x) / 2
1/2
f is an even function


25. The Product Rule

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26. Average speed
f(x) g'(x) + g(x) f'(x)
1
1 / sin x
?s/?t


27. cos 3p/2
2pr
0
(ln a) a^x
(1 + cos 2x) / 2


28. ln (mn)
Derivative of position at a point
S = 4 pi r^2
(1 + cos 2x) / 2
ln m + ln n


29. ln e
1 / tan x = cos x / sin x
v3/2
v2/2
1


30. csc x
1
-csc² x
1 / sin x
v2/2


31. Power Rule for Derivatives
d/dx[x^n]=nx^(n-1)
-sin x
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(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


32. Critical number

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33. Intermediate Value Theorem
(1 - cos 2x) / 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.
?s/?t
Slope of a function at a point/slope of the tangent line to a function at a point


34. d/dx[e^x]
pr²h
e^x
1
v2/2


35. Guidelines for implicit differentiation

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36. Derivative of a constant
1
?s/?t
csc²x
d/dx[c] = 0


37. How to get from precalculus to calculus
Limits
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
S = 4 pi r^2
V = 4/3 pi r^3


38. cos(2x)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
d/dx[cf(x)] = c f'(x)
cos²x - sin²x
v3/2


39. Area of an equilateral triangle
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)
pr²h
v3s² / 4
1 / tan x = cos x / sin x


40. sin p/4
1/(|x|v(x²-1))
v3/2
v3/2
v2/2


41. cos 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
f is an even function
1
V = 4/3 pi r^3


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

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43. sin 0
2 sin x cos x
0
sec x tan x
(1 + cos 2x) / 2


44. cos²x
-1/v(1-x²)
d/dx[cf(x)] = c f'(x)
(1 + cos 2x) / 2
u'/u - u > 0


45. Guidelines for solving related rates problems
1
sec x tan 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
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)


46. cos p/4
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
1 / cos x
v2/2
Derivative of position at a point


47. cot x
1 / tan x = cos x / sin x
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]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


48. sin(2x)
Derivative of position at a point
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
e^x
2 sin x cos x


49. Surface Area of a Sphere
1 / cos x
cos x
S = 4 pi r^2
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]


50. d/dx[x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
sin x / cos x
pr²
1