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[arccsc x]
-1/(|x|v(x²-1))
x values where f'(x) is zero or undefined.
ln m - ln n
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


2. sin p/6
0
1
V = 4/3 pi r^3
1/2


3. 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.
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'
-csc² x
ln m - ln n


4. ln (mn)
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
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
ln m + ln n
Derivative of position at a point


5. d/dx[a^x]
d/dx[x^n]=nx^(n-1)
0
(ln a) a^x
1/2


6. cos p/3
1/2
-csc² x
u'/((ln a) u)
x values where f'(x) is zero or undefined.


7. ln 1
u' (ln a) a^u
?s/?t
1
0


8. d/dx[arccos x]
?s/?t
-1/v(1-x²)
0
1/x - x>0


9. Continuity on a closed interval - [a -b]
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)
sec x tan x
sec² x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)


10. The limit as x approaches 0 of (1 - cos x) / x
u'/u - u > 0
(ln a) a^x
0
-csc x cot x


11. Area of an equilateral triangle
f(x) g'(x) + g(x) f'(x)
v3s² / 4
1
f'(g(x))g'(x)


12. ln (m/n)
cos x
ln m - ln n
pr²
Limits


13. cos p/6
v3/2
-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)
cos x


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


15. Average speed
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
sin x / cos x
?s/?t
1


16. d/dx[e^x]
1/v(1-x²)
1 / cos x
1/x - x>0
e^x


17. sin(2x)
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)
sec x tan x
2 sin x cos x
sin x / cos x


18. Critical number

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19. cos p/4
csc²x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v2/2
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)


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


21. Indeterminate form
0/0
e^x
ln m + ln n
d/dx[c] = 0


22. d/dx[cot x]
-1/v(1-x²)
sin x / cos x
x values where f'(x) is zero or undefined.
-csc² x


23. cos²x
u' (ln a) a^u
(1 + cos 2x) / 2
1/v(1-x²)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].


24. If f(-x) = -f(x)
f is an odd function
ln m + ln n
sec x tan x
Derivative of Position - s'(t)


25. csc x
1 / sin 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.
0
Slope of a function at a point/slope of the tangent line to a function at a point


26. Sum and Difference Rules for Derivatives

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27. cos p/2
e^x
0
pr²h
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


28. Volume of a Sphere
-1
V = 4/3 pi r^3
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
Slope of a function at a point/slope of the tangent line to a function at a point


29. d/dx[sec x]
sec x tan x
1 / cos x
1
f(x) g'(x) + g(x) f'(x)


30. d/dx[arcsin x]
?s/?t
1/v(1-x²)
u'/u - u > 0
Derivative of Position - s'(t)


31. Instantaneous velocity
(1 + cos 2x) / 2
ln m - ln n
1/(1+x²)
Derivative of position at a point


32. d/dx[sin x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
cos x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


33. cos 3p/2
0
v2/2
Limits
V = 4/3 pi r^3


34. sin p/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).
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1
Limits


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

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36. d/dx[ln u]

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37. cos²x + sin²x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
0/0
?s/?t
1


38. ln mn
(ln a) a^x
n ln m
1 / sin x
v3/2


39. 1 + tan²x
sec²x
v3s² / 4
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
cos x


40. Volume of a right circular cylinder
-1/(|x|v(x²-1))
u' e^u
-1
pr²h


41. sin p/4
v2/2
cos²x - sin²x
-1/(|x|v(x²-1))
0


42. d/dx[log_a x]
1/v(1-x²)
Slope of a function at a point/slope of the tangent line to a function at a point
1/((ln a) x)
0


43. Continuity on an open interval - (a -b)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
sin x / cos x
1/x - x>0
sec² x


44. Circumference of a circle
2pr
-1
0
Derivative of Position - s'(t)


45. If f(-x) = f(x)
f is an even function
csc²x
-sin x
0


46. tan x
0
v2/2
sin x / cos x
S = 4 pi r^2


47. sin p/3
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v3/2
0/0
d/dx[cf(x)] = c f'(x)


48. Limit Definition of a Derivative

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49. d/dx[arccot x]
-1/(1+x²)
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
sec x tan 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


50. The limit as x approaches 0 of sin x / x
d/dx[cf(x)] = c f'(x)
1
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
-csc² x