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AP Calculus Formulas
Start Test
Study First
Subjects
:
math
,
ap
,
calculus
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[ f(x) g(x) ]
2. Continuity & differentiability
0
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v3s² / 4
3. sin²x
u'/((ln a) u)
Limits
(1 - cos 2x) / 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).
4. Volume of a Sphere
1/2
Derivative of position at a point
v2/2
V = 4/3 pi r^3
5. d/dx[arcsec x]
1/(|x|v(x²-1))
pr²h/3
Derivative of Position - s'(t)
1
6. Continuity on an open interval - (a -b)
v3/2
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
S = 4 pi r^2
-1
7. Instantaneous velocity
1 / cos x
Derivative of position at a point
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
8. d/dx[e^x]
e^x
-csc x cot 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
pr²
9. d/dx[cos x]
-1/(|x|v(x²-1))
e^x
f is an even function
-sin x
10. cot x
u'/u - u > 0
sin x / cos x
-sin x
1 / tan x = cos x / sin x
11. sin p/4
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'
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
0
v2/2
12. sin 0
(ln a) a^x
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
13. How to get from precalculus to calculus
-csc x cot x
f(x) g'(x) + g(x) f'(x)
Limits
sec²x
14. cos p/2
csc²x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
-1
0
15. cos 3p/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
-1/(|x|v(x²-1))
1/(1+x²)
0
16. Alternate Limit Definition of a derivative
17. d/dx[ln u]
18. sin p/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
1/x - x>0
v3/2
19. d/dx[cot x]
Slope of a function at a point/slope of the tangent line to a function at a point
e^x
-csc² x
1/x - x>0
20. Area of an equilateral triangle
v3s² / 4
Limits
-1
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
21. sin p/3
v3/2
n ln m
1
1
22. d/dx[sin x]
cos x
d/dx[c] = 0
v3s² / 4
-csc² x
23. Average speed
u'/((ln a) u)
f(x) g'(x) + g(x) f'(x)
?s/?t
Derivative of Position - s'(t)
24. sec x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1 / cos x
1 / sin x
25. cos p/4
1/x - x>0
V = 4/3 pi r^3
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'
v2/2
26. cos p
(1 + cos 2x) / 2
-1
(1 - cos 2x) / 2
V = 4/3 pi r^3
27. d/dx[ln x]
1/v(1-x²)
sec²x
1/x - x>0
d/dx[cf(x)] = c f'(x)
28. d/dx[log_a x]
u' (ln a) a^u
1/2
1/((ln a) x)
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
29. d/dx[x]
1
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
u'/((ln a) u)
1 / sin x
30. Constant Multiple Rule for Derivatives
31. csc x
-1/(1+x²)
f is an even function
f'(g(x))g'(x)
1 / sin x
32. sin p
pr²h/3
0
(ln a) a^x
-sin x
33. Indeterminate form
n ln m
1
0/0
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
34. sin p/6
Derivative of position at a point
1/2
-csc² x
d/dx[cf(x)] = c f'(x)
35. ln e
?s/?t
1
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
-1
36. ln 1
-csc² 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).
sec²x
0
37. d/dx[ f(x) / g(x) ]
38. Circumference of a circle
0
-1
2pr
e^x
39. Surface Area of a Sphere
(ln a) a^x
S = 4 pi r^2
0
-csc² x
40. Power Rule for Derivatives
d/dx[x^n]=nx^(n-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.
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 / tan x = cos x / sin x
41. d/dx[sec x]
-sin x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
u'/u - u > 0
sec x tan x
42. cos p/6
v3/2
0
sec x tan x
-1/v(1-x²)
43. d/dx[tan x]
-1/(1+x²)
u'/u - u > 0
sec² x
1
44. cos²x + sin²x
v3/2
Derivative of Position - s'(t)
1/2
1
45. The Quotient Rule
46. The limit as x approaches 0 of (1 - cos x) / x
0
-1/(1+x²)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
x values where f'(x) is zero or undefined.
47. Rolle's Theorem
48. The Product Rule
49. cos p/3
S = 4 pi r^2
1/2
n ln m
0/0
50. d/dx[arccot x]
Limits
pr²h
1 / cos x
-1/(1+x²)
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