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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 re-enforces your understanding as you take the test each time.
1. ln mn
0
1/x - x>0
n ln m
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
2. Average speed
v2/2
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
u' (ln a) a^u
?s/?t
3. d/dx[x]
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1
4. d/dx[a^u]
5. d/dx[ln u]
6. The limit as x approaches 0 of (1 - cos x) / x
0/0
0
pr²h
-sin x
7. 1 + cot²x
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'
d/dx[c] = 0
csc²x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
8. cot x
1 / tan x = cos x / sin x
1 / sin x
pr²h
sec x tan x
9. sin p/2
0
2 sin x cos x
1
-csc x cot x
10. Position function of a falling object (with acceleration in ft/s²)
v2/2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
2pr
1/(1+x²)
11. Chain Rule: d/dx[f(g(x))] =
12. Continuity on an open interval - (a -b)
csc²x
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) a^u
e^x
13. cos p/6
S = 4 pi r^2
v3/2
u' e^u
(1 - cos 2x) / 2
14. Continuity on a closed interval - [a -b]
-1/(|x|v(x²-1))
1/x - x>0
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)
0
15. Intermediate Value Theorem
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)
1/((ln a) x)
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 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.
16. Continuity & differentiability
u'/u - u > 0
csc²x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v2/2
17. sin 3p/2
pr²
-1/(|x|v(x²-1))
0
-1
18. cos p
(ln a) a^x
Slope of a function at a point/slope of the tangent line to a function at a point
-1
d/dx[cf(x)] = c f'(x)
19. d/dx[ f(x) g(x) ]
20. Position function of a falling object (with acceleration in m/s²)
-csc² 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
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u' (ln a) a^u
21. Surface Area of a Sphere
-1
f'(g(x))g'(x)
-sin x
S = 4 pi r^2
22. d/dx[sec x]
v3s² / 4
1/(1+x²)
sec x tan x
u' e^u
23. Derivative of a constant
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
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'
d/dx[c] = 0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
24. Sum and Difference Rules for Derivatives
25. cos²x + sin²x
1/x - x>0
1
S = 4 pi r^2
1/2
26. Constant Multiple Rule for Derivatives
27. If f(-x) = f(x)
1/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/v(1-x²)
f is an even function
28. Indeterminate form
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
-1/(1+x²)
0
0/0
29. sin p/6
V = 4/3 pi r^3
-1/(|x|v(x²-1))
1
1/2
30. Guidelines for implicit differentiation
31. tan x
?s/?t
sin x / cos x
u'/u - u > 0
-csc² x
32. 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)
x values where f'(x) is zero or undefined.
Limits
2 sin x cos x
33. If f(-x) = -f(x)
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'
pr²
pr²h
f is an odd function
34. cos(2x)
pr²h
cos²x - sin²x
x values where f'(x) is zero or undefined.
1/2
35. Circumference of a circle
2pr
Limits
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
36. d/dx[arccos x]
v3s² / 4
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.
sec x tan x
-1/v(1-x²)
37. Limit Definition of a Derivative
38. 1 + tan²x
sec²x
1 / sin x
1
1/x - x>0
39. csc x
pr²
1/(1+x²)
1 / sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
40. d/dx[arcsin x]
1/v(1-x²)
-csc x cot x
1/((ln a) x)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
41. cos p/2
1/2
-1
0
Slope of a function at a point/slope of the tangent line to a function at a point
42. Instantaneous velocity
-csc x cot x
Derivative of position at a point
cos x
u'/((ln a) u)
43. d/dx[arccsc x]
Derivative of position at a point
-1/(|x|v(x²-1))
(1 + cos 2x) / 2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
44. sin 0
v2/2
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
u'/u - u > 0
45. cos p/3
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/v(1-x²)
1/2
0
46. 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
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
1/((ln a) x)
47. Critical number
48. Volume of a cone
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
Limits
pr²h/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).
49. d/dx[csc x]
-csc x cot x
0
cos x
1
50. sin p
2 sin x cos x
f is an even function
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
0