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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. Derivative of a constant
pr²h/3
0
d/dx[c] = 0
1
2. Indeterminate form
V = 4/3 pi r^3
0/0
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
2pr
3. Volume of a right circular cylinder
0
Derivative of position at a point
f is an even function
pr²h
4. How to get from precalculus to calculus
0
pr²h
Limits
v2/2
5. d/dx[arccot x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1
1
1/(1+x²)
6. Guidelines for implicit differentiation
7. Rolle's Theorem
8. d/dx[ f(x) / g(x) ]
9. d/dx[log_a u]
10. sin p/6
d/dx[x^n]=nx^(n1)
1/2
0
Limits
11. The limit as x approaches 0 of (1  cos x) / x
(ln a) a^x
1/(xv(x²1))
0
ln m  ln n
12. cos p/6
Derivative of position at a point
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
v3/2
V = 4/3 pi r^3
13. Continuity on an open interval  (a b)
v2/2
f(x) is continuous if for every point on the interval (a b) the conditions for continuity at a point are satisfied.
Slope of a function at a point/slope of the tangent line to a function at a point
1/(xv(x²1))
14. The Product Rule
15. sec x
ln m  ln n
1 / cos x
1
d/dx[c] = 0
16. If f(x) = f(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)
0
f is an odd function
x values where f'(x) is zero or undefined.
17. Sum and Difference Rules for Derivatives
18. Chain Rule: d/dx[f(g(x))] =
19. sin p/2
n ln m
d/dx[cf(x)] = c f'(x)
1
v3s² / 4
20. cos(2x)
f is an even function
V = 4/3 pi r^3
s(t) = 16t²+ v0t + s0  v0 = initial velocity  s0 = initial height
cos²x  sin²x
21. d/dx[log_a x]
1/((ln a) x)
0
S = 4 pi r^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
22. d/dx[a^x]
(ln a) a^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).
pr²h/3
n ln m
23. 1 + cot²x
(1 + cos 2x) / 2
u'/u  u > 0
csc²x
Slope of a function at a point/slope of the tangent line to a function at a point
24. Derivative of an inverse (if g(x) is the inverse of f(x))
25. Critical number
26. ln 1
0
Limits
1/v(1x²)
pr²h
27. Area of an equilateral triangle
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
v3s² / 4
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
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'
28. sin p
f'(x) = lim as ?x ? 0 of [ f(x + ?x)  f(x) ] / ?x
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.
1
29. ln mn
f is an even function
n ln m
pr²h/3
1/((ln a) x)
30. sin²x
(1  cos 2x) / 2
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
pr²h
1/(xv(x²1))
31. Derivative
Slope of a function at a point/slope of the tangent line to a function at a point
ln m  ln n
v2/2
v3s² / 4
32. cos p
1
v3s² / 4
f'(g(x))g'(x)
s(t) = 16t²+ v0t + s0  v0 = initial velocity  s0 = initial height
33. sin p/3
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
0/0
v3/2
34. d/dx[arccos x]
S = 4 pi r^2
d/dx[c] = 0
0
1/v(1x²)
35. d/dx[arccsc x]
1
1/(xv(x²1))
1/(1+x²)
v3/2
36. d/dx[sec x]
sec x tan x
f(x) g'(x) + g(x) f'(x)
Derivative of position at a point
v3/2
37. Mean Value Theorem
38. d/dx[a^u]
39. Constant Multiple Rule for Derivatives
40. csc x
1/v(1x²)
1 / sin x
f(x) is continuous if for every point on the interval (a b) the conditions for continuity at a point are satisfied.
f is an odd function
41. sin 3p/2
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)
1
Derivative of Position  s'(t)
0
42. Guidelines for solving related rates problems
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
csc²x
v3/2
43. d/dx[arctan x]
1/(1+x²)
0/0
s(t) = 4.9t²+ v0t + s0  v0 = initial velocity  s0 = initial height
1
44. The limit as x approaches 0 of sin x / x
1
(1 + cos 2x) / 2
If two functions  f and g  are differentiable  then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
Derivative of Position  s'(t)
45. cos p/3
1/(1+x²)
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
1/2
e^x
46. Volume of a Sphere
Differentiability implies continuity  but continuity does not necessarily imply differentiability.
V = 4/3 pi r^3
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
47. cos 3p/2
1/(xv(x²1))
sec²x
x values where f'(x) is zero or undefined.
0
48. d/dx[cos x]
s(t) = 16t²+ v0t + s0  v0 = initial velocity  s0 = initial height
0
sec² x
sin x
49. cos p/2
0
1/v(1x²)
sec²x
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'
50. Area of a circle
1/(1+x²)
cos x
1
pr²