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. Indeterminate form
0/0
pr²
f is an odd function
v3s² / 4


2. d/dx[arcsec x]
f'(g(x))g'(x)
1/(|x|v(x²-1))
0
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


3. d/dx[arcsin x]
v3s² / 4
?s/?t
0
1/v(1-x²)


4. Surface Area of a Sphere
1/v(1-x²)
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.
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
S = 4 pi r^2


5. Position function of a falling object (with acceleration in ft/s²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3/2
pr²
-1/(|x|v(x²-1))


6. sec x
f'(g(x))g'(x)
0
1 / cos x
1/v(1-x²)


7. d/dx[arccos x]
-1/v(1-x²)
pr²h/3
d/dx[cf(x)] = c f'(x)
Slope of a function at a point/slope of the tangent line to a function at a point


8. Rolle's Theorem

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9. d/dx[cos x]
f is an odd function
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
-sin x
Slope of a function at a point/slope of the tangent line to a function at a point


10. Limit Definition of a Derivative

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11. Chain Rule: d/dx[f(g(x))] =

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12. d/dx[tan x]
sin x / cos x
sec² x
V = 4/3 pi r^3
(ln a) a^x


13. Volume of a right circular cylinder
1/2
1/2
pr²h
d/dx[c] = 0


14. Mean Value Theorem

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15. Area of an equilateral triangle
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
v3s² / 4
v3/2
v2/2


16. ln (mn)
v3/2
-1/(1+x²)
1 / tan x = cos x / sin x
ln m + ln n


17. sin p/2
pr²
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
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height


18. Guidelines for solving related rates problems
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. 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
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-csc² x


19. d/dx[log_a x]
0
1/2
1/((ln a) x)
-csc x cot x


20. d/dx[e^u]

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21. sin 3p/2
-1/(1+x²)
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'
-1
pr²h/3


22. sin p
0
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Limits
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).


23. d/dx[arctan x]
f is an even function
0
1/(1+x²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)


24. d/dx[arccsc x]
-1/(|x|v(x²-1))
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/(1+x²)


25. d/dx[sin x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
ln m + ln n
cos x


26. Instantaneous velocity
v2/2
Derivative of position at a point
1/((ln a) x)
n ln m


27. ln e
pr²h/3
S = 4 pi r^2
d/dx[cf(x)] = c f'(x)
1


28. Derivative of an inverse (if g(x) is the inverse of f(x))

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29. d/dx[log_a u]

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30. ln mn
0
d/dx[x^n]=nx^(n-1)
n ln m
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)


31. d/dx[sec x]
u' (ln a) a^u
e^x
sec x tan x
?s/?t


32. cos 3p/2
f'(g(x))g'(x)
2 sin x cos x
0
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. cos p
1/2
-1
0
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x


34. Volume of a cone
sec² x
u' e^u
0/0
pr²h/3


35. Position function of a falling object (with acceleration in m/s²)
(ln a) a^x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m + ln n
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)


36. Area of a circle
csc²x
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.
1/(|x|v(x²-1))


37. Continuity & differentiability
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v2/2
-1
cos²x - sin²x


38. ln 1
1/(|x|v(x²-1))
-1
csc²x
0


39. cos(2x)
cos²x - sin²x
v3/2
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1/((ln a) x)


40. cos p/2
0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
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)
d/dx[c] = 0


41. Average speed
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/v(1-x²)
(1 + cos 2x) / 2
?s/?t


42. cos p/3
v3/2
1/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.


43. Circumference of a circle
2pr
u'/((ln a) u)
Slope of a function at a point/slope of the tangent line to a function at a point
(1 - cos 2x) / 2


44. Constant Multiple Rule for Derivatives

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45. If f(-x) = -f(x)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - 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)
f is an odd function
0


46. Continuity at a point (x = c)
pr²
Limits
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)
csc²x


47. cot x
1 / tan x = cos x / sin x
v3/2
-1
v3s² / 4


48. Volume of a Sphere
d/dx[cf(x)] = c f'(x)
1
V = 4/3 pi r^3
1 / sin x


49. 1 + tan²x
2 sin x cos x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
sec²x
u'/u - u > 0


50. ln (m/n)
sec² x
ln m - ln n
1/v(1-x²)
f is an even function