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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. Guidelines for implicit differentiation
2. sin p
0
-1/v(1-x²)
v3s² / 4
Limits
3. cos p/6
v3/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u'/u - u > 0
x values where f'(x) is zero or undefined.
4. d/dx[arctan x]
u' e^u
ln m - ln n
1/(1+x²)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
5. Intermediate Value Theorem
f'(g(x))g'(x)
1 / sin x
1 / cos x
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.
6. Position function of a falling object (with acceleration in m/s²)
Derivative of position at a point
-csc² x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
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).
7. Extreme Value Theorem
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
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/(|x|v(x²-1))
1 / cos x
8. d/dx[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
1
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/v(1-x²)
9. Guidelines for solving related rates problems
2 sin x cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
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
10. The limit as x approaches 0 of sin x / x
1
V = 4/3 pi r^3
Slope of a function at a point/slope of the tangent line to a function at a point
-1/v(1-x²)
11. 1 + tan²x
sec²x
-1/v(1-x²)
1
sin x / cos x
12. d/dx[ln u]
13. Volume of a Sphere
V = 4/3 pi r^3
1/v(1-x²)
sec x tan x
1
14. sin p/4
d/dx[c] = 0
v2/2
V = 4/3 pi r^3
-1/(1+x²)
15. d/dx[a^x]
x values where f'(x) is zero or undefined.
v3s² / 4
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
(ln a) a^x
16. d/dx[cos x]
-sin x
1
-csc² x
(1 + cos 2x) / 2
17. ln 1
u'/((ln a) u)
0
u'/u - u > 0
1 / tan x = cos x / sin x
18. sin p/3
v3/2
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
ln m + ln n
1
19. d/dx[arcsin x]
1/v(1-x²)
f is an even function
sin x / cos x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
20. Constant Multiple Rule for Derivatives
21. d/dx[a^u]
22. ln (mn)
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
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)
ln m + ln n
23. Critical number
24. sin(2x)
1
(1 + cos 2x) / 2
2 sin x cos x
u'/((ln a) u)
25. sec x
1 / cos x
d/dx[c] = 0
f is an odd function
0
26. d/dx[arccsc x]
1/2
-1/(|x|v(x²-1))
1
d/dx[cf(x)] = c f'(x)
27. d/dx[tan x]
2pr
ln m + ln n
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)
sec² x
28. d/dx[csc x]
-1/v(1-x²)
sec x tan x
d/dx[x^n]=nx^(n-1)
-csc x cot x
29. cos p/4
-1/(|x|v(x²-1))
v2/2
2pr
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)
30. The limit as x approaches 0 of (1 - cos x) / x
v3/2
cos²x - sin²x
0
1/((ln a) x)
31. d/dx[arcsec x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
V = 4/3 pi r^3
0
1/(|x|v(x²-1))
32. d/dx[e^u]
33. Surface Area of a Sphere
pr²
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.
S = 4 pi r^2
v3s² / 4
34. Circumference of a circle
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)
2pr
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1
35. Position function of a falling object (with acceleration in ft/s²)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f is an even function
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec²x
36. Limit Definition of a Derivative
37. Rolle's Theorem
38. cos(2x)
cos²x - sin²x
Derivative of position at a point
sec²x
f(x) g'(x) + g(x) f'(x)
39. sin p/2
pr²h/3
-csc x cot x
0
1
40. Chain Rule: d/dx[f(g(x))] =
41. The Product Rule
42. Volume of a right circular cylinder
v3/2
-csc x cot x
pr²h
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
43. cos p
-1
u'/u - u > 0
-1/(1+x²)
v3s² / 4
44. Continuity at a point (x = c)
1
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
-csc² 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)
45. Continuity on an open interval - (a -b)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
pr²h
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
cos²x - sin²x
46. ln (m/n)
S = 4 pi r^2
1/2
ln m - ln n
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
47. cos p/2
n ln m
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m - ln n
0
48. ln mn
1
n ln m
-1/v(1-x²)
0
49. d/dx[log_a x]
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/((ln a) x)
1
-1/(|x|v(x²-1))
50. cos 3p/2
cos x
0
Limits
v2/2