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. Continuity on an open interval - (a -b)
pr²h
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
v3/2


2. The Product Rule

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

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4. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
Derivative of position at a point
Slope of a function at a point/slope of the tangent line to a function at a point
f is an odd function


5. cos(2x)
-sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
cos²x - sin²x


6. Derivative
sin x / cos x
v3/2
1
Slope of a function at a point/slope of the tangent line to a function at a point


7. d/dx[ln x]
0
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. 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 - x>0


8. Alternate Limit Definition of a derivative

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9. d/dx[a^u]

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10. Instantaneous velocity
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)
cos x
Derivative of position at a point
(1 - cos 2x) / 2


11. Volume of a Sphere
V = 4/3 pi r^3
u'/((ln a) u)
-sin x
pr²


12. ln (m/n)
ln m - ln n
(1 + cos 2x) / 2
1
f'(g(x))g'(x)


13. d/dx[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)
x values where f'(x) is zero or undefined.
n ln m
1


14. 1 + tan²x
sec² x
0
sec²x
Derivative of Position - s'(t)


15. Sum and Difference Rules for Derivatives

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16. How to get from precalculus to calculus
Limits
2 sin x cos x
-csc x cot x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


17. The Quotient Rule

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18. d/dx[log_a u]

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19. Mean Value Theorem

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


21. Derivative of a constant
1
d/dx[c] = 0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1/((ln a) x)


22. Intermediate Value Theorem
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.
v2/2
0
1/2


23. d/dx[tan 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)
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).
Derivative of Position - s'(t)
sec² x


24. Continuity at a point (x = c)
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
f is an even function
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).


25. ln 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)
0
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
v2/2


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

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27. cos²x + sin²x
1
cos x
(1 + cos 2x) / 2
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0


28. sec x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
1
1
1 / cos x


29. Guidelines for implicit differentiation

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30. d/dx[arcsec x]
1/(|x|v(x²-1))
d/dx[cf(x)] = c f'(x)
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)


31. d/dx[arccot x]
-1/(1+x²)
d/dx[c] = 0
0
n ln m


32. sin p/3
sec² x
u' (ln a) a^u
v3/2
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).


33. If f(-x) = -f(x)
S = 4 pi r^2
e^x
sec² x
f is an odd function


34. Continuity on a closed interval - [a -b]
1
d/dx[cf(x)] = c f'(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)
csc²x


35. The limit as x approaches 0 of (1 - cos x) / x
u'/((ln a) u)
-1/v(1-x²)
0
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


36. cos 0
1
pr²h/3
2pr
u'/((ln a) u)


37. sin 3p/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
cos x
-1
1


38. Circumference of a circle
2pr
sec x tan 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.
1 / cos x


39. Volume of a right circular cylinder
pr²h
cos x
0
d/dx[x^n]=nx^(n-1)


40. d/dx[e^x]
e^x
u' e^u
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
n ln m


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

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42. sin(2x)
0/0
1/((ln a) x)
2 sin x cos x
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)


43. d/dx[ln u]

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44. d/dx[cot x]
-csc² x
0
f is an odd function
-1/(|x|v(x²-1))


45. d/dx[arccos x]
0/0
-1/v(1-x²)
1
Limits


46. If f(-x) = f(x)
1
f is an even 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.
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


47. cos p/2
v2/2
0
f(x) g'(x) + g(x) f'(x)
1/v(1-x²)


48. 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[x^n]=nx^(n-1)
-1/(|x|v(x²-1))
u'/((ln a) u)


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

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50. csc x
1
1 / sin x
1/2
-csc² x