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Test your basic knowledge 
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 reenforces 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/(xv(x²1))
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/(xv(x²1))
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/(xv(x²1))
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(1x²)
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^(n1)
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(1x²)
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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