| Review 1 | Review 2 | Review 3 | Review 4 | Review 5 |
|---|---|---|---|---|
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Concave up
The tangent line is an underestimate if f(x) is
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Concave down
The tangent line is an overestimate if f(x) is ____
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-f(x)
-f(g(x)) x g'(x)
FTC Part 2:
-d/dx integral from a to x of f(t) dt= -d/dx integral from a to g(x) of f(t) dt= |
V= (π) integral from a to b of ((outer-axis)^2 - (inner-axis)^2) dx
Volume by Revolution=
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-A jump discontinuity
-An infinite discontinuity
What is a non-removable discontinuity?
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Continuous
If a function is differentiable, then it is ______
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A hole in the graph where the limit as x approaches a of f(x) doesn't equal f(a)
What is a removable discontinuity?
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(1/(b-a)) integral from a to b of f(x) dx
Average Value=
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((p(b)-p(a)) / (b-a))
or (1/(b-a)) integral from a to b of V(t) dt
Average Velocity
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A line that's perpendicular to the tangent line. The slop of the normal line is negative reciprocal of the slope of the tangent line (the derivative).
Normal Line
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f(x) has a relative minimum at x=a
If f'(a)=0 and f'(x) changes from negative to positive at x=a, then ____
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A point where f'(x)=0 or f'= undefined
Critical Point
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((f(b)-f(a)) / (b-a)) } slope of the secant line
What is Average Rate of Change?
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V=1/2 integral from a to b (top function-bottom function)^2 dx
Cross-Section (Isosceles Right Triangle with leg on base)
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1st- Find critical points: where f'(x)=0
2nd- Evaluate f(x) at the critical point and endpoints 3rd- Biggest Value= Absolute maximum Lowest Value= Absolute minimum
How do you find absolute extremes?
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f(x) has a relative maximum at x=a
If f'(a)=0 and f'(x) changes from positive to negative at x=a, then ____
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-not continuous
-vertical tangent line -sharp corner -cusp
What functions are non-differentiable?
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If a is a critical point and if
-f(a) > 0 --> x --> x=a is a relative minimum -f(a) < 0 --> x --> x=a is a relative maximum -f(a) = 0 --> inconclusive
2nd derivative test
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integral from a to b (f(x)-g(x)) dx (top-bottom)
or integral from a to b (f(y)-g(y)) dy (right-left)
Area between two curves =
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1.limit as x approaches a of f(x) exists
2.f(a) exists 3.limit as x approaches a of f(x) = f(a)
A function is continuous at x=a if _____
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The area bounded by f(x) and the x-axis from x=a to x=b
What does a definite integral represent?
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f" < 0
or f' is increasing
f(x) is concave down if ____
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f"=0 or undefined and f" changes sign
or f' changes direction
f(x) has a point of inflection if ____
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V=integral from a to b A(x)^2 dx (Area of cross section)
V=integral from a to b A(y)^2 dy (perpendicular to y-axis)
Volume by Cross-Section (Square)
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limit as n approaches infinite n∑i=1 f(a+change in x times c) x (change in x) where change in x= ((b-a) /n)
Definition of Definite Integral
Integral from a to b of f(x) dx= |
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