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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. Sum and Difference Rules for Derivatives
2. The Product Rule
3. d/dx[arcsin x]
1/2
1
1/v(1x²)
1 / cos x
4. If f(x) = f(x)
sec x tan x
f is an odd function
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)
sec² x
5. d/dx[arccos x]
1/v(1x²)
x values where f'(x) is zero or undefined.
u' e^u
1/v(1x²)
6. The Quotient Rule
7. d/dx[arccsc x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1
1/(xv(x²1))
If two functions  f and g  are differentiable  then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
8. ln e
0
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
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
9. sin²x
(1  cos 2x) / 2
?s/?t
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).
1/2
10. Derivative
d/dx[cf(x)] = c f'(x)
1
Slope of a function at a point/slope of the tangent line to a function at a point
ln m  ln n
11. d/dx[e^u]
12. d/dx[arctan 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
1/(1+x²)
n ln m
1/(xv(x²1))
13. cos 3p/2
1/((ln a) x)
1 / sin x
Limits
0
14. Volume of a Sphere
d/dx[cf(x)] = c f'(x)
V = 4/3 pi r^3
u'/((ln a) u)
u' e^u
15. csc x
?s/?t
1 / sin x
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
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
16. ln (m/n)
s(t) = 16t²+ v0t + s0  v0 = initial velocity  s0 = initial height
1
1/v(1x²)
ln m  ln n
17. cos²x + sin²x
1/2
If two functions  f and g  are differentiable  then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
cos²x  sin²x
1
18. d/dx[ f(x) g(x) ]
19. Indeterminate form
pr²h/3
0/0
0
1/2
20. d/dx[tan x]
f is an even function
d/dx[cf(x)] = c f'(x)
sec² x
Slope of a function at a point/slope of the tangent line to a function at a point
21. d/dx[ln x]
1/x  x>0
1
1
1
22. Position function of a falling object (with acceleration in m/s²)
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
pr²h/3
v2/2
ln m + ln n
23. d/dx[cot x]
u' e^u
0
1
csc² x
24. d/dx[log_a u]
25. cos²x
x values where f'(x) is zero or undefined.
1/(xv(x²1))
0
(1 + cos 2x) / 2
26. Extreme Value Theorem
pr²h/3
(ln a) a^x
If f is continuous on the closed interval [a b] then it must have both a minimum and maximum on [a b].
1/v(1x²)
27. Continuity at a point (x = c)
Derivative of position at a point
sec² 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)
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)
28. ln (mn)
1/x  x>0
f is an odd function
ln m + ln n
Derivative of position at a point
29. sec x
v2/2
1 / cos x
e^x
sec x tan x
30. How to get from precalculus to calculus
1/(xv(x²1))
f is an even function
Limits
u' e^u
31. d/dx[e^x]
1
f(x) is continuous if for every point on the interval (a b) the conditions for continuity at a point are satisfied.
e^x
f'(g(x))g'(x)
32. d/dx[log_a x]
1/((ln a) x)
1/(1+x²)
1
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)
33. Area of an equilateral triangle
ln m  ln n
0
v3s² / 4
1
34. sin 3p/2
csc²x
1
2 sin x cos x
(1  cos 2x) / 2
35. sin 0
v2/2
0
sec² x
1/((ln a) x)
36. d/dx[arccot x]
sec x tan x
1/(1+x²)
1/(1+x²)
f(x) is continuous if for every point on the interval (a b) the conditions for continuity at a point are satisfied.
37. Continuity on a closed interval  [a b]
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)
v2/2
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
38. sin p/6
1/2
u'/u  u > 0
1/v(1x²)
1
39. Derivative of a constant
1
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
d/dx[c] = 0
1/(1+x²)
40. 1 + tan²x
pr²h/3
u' (ln a) a^u
sec²x
1/x  x>0
41. ln 1
v2/2
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.
sin x / cos x
0
42. tan x
sin x / cos x
x values where f'(x) is zero or undefined.
Slope of a function at a point/slope of the tangent line to a function at a point
sin x
43. d/dx[cos x]
1/(1+x²)
2 sin x cos x
csc x cot x
sin x
44. sin p/2
1
sin x
csc²x
v3s² / 4
45. d/dx[ln u]
46. cos p/3
1/2
v3s² / 4
(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.
47. Guidelines for implicit differentiation
48. d/dx[a^u]
49. Continuity & differentiability
If f is continuous on the closed interval [a b] then it must have both a minimum and maximum on [a b].
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
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
1/2
50. Intermediate Value Theorem
1/v(1x²)
u'/u  u > 0
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.
g'(x) = 1/f'(g(x))  f'(g(x)) cannot = 0