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Definition
~ solve for x and set equal to zero
~ Watch for Domain Issues
~ On calc. graph and find zeros
~ Use quadratic equation if needed |
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Term
| Find equation of the line tangent to f(x) at (a,b). |
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Definition
y-y1=m(x-x1)
Slope: take derivative and plug in x value |
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Term
| Find equation of the line normal (perpendicular) to f(x) at (a,b). |
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Definition
| ~ Like tangent line. When you find "m" do the negative reciprocal. |
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Term
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Definition
~ Symmetric along y-axis
~ plug in -x to make sure nothing changes |
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Term
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Definition
~ Symmetric along the origin
~ plug in -x and -y to make sure nothing changes |
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Term
| Find the interval where the f(x) is increasing. |
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Definition
| ~ 1st derivative: critical #'s on number line (f' line) |
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Term
| Find interval where the slope of f(x) is increasing. |
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Definition
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Term
| Find the minimum value (absolute) value of a function. |
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Definition
~ plug endpoints and critical numbers into original
~ find smallest y value |
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Term
| Find the minimum slope of a function. |
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Definition
| ~ Plug endpoints and P.O.I. into the derivative. |
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Term
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Definition
~ 1st derivative and set equal to zero.
~ Watch for Domain issues. |
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Term
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Definition
~ 2nd derivative and set equal to zero.
~ Where there is a change in concavity. |
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Term
Show that lim f(x) exists.
x-->a |
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Definition
lim f(x)= lim f(x)
x->a+ x->a- |
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Term
| Show that f(x) is continuous. |
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Definition
~ lim f(x) exists
x->a
~ f(a) exists
~ lim f(x)= f(a)
x->a |
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Term
| Find vertical asymptotes of f(x). |
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Definition
~ Set denominator equal to zero.
~ Only happens when there is an x in the denominator. |
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Term
| Find horizontal asymptotes of f(x). |
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Definition
~ Plug in infinity for x.
~ Only happens when there is an x in the denominator. |
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Term
| Find the average rate of change of f(x) on [a,b]. |
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Definition
~ Slope between two points.
~ y2-y1/x2-x1
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Term
| Find instantaneous rate of change on f(x) at a. |
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Definition
~ Velocity at one point.
~ Find f' |
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Term
| Find the average value of f(x) on [a,b]. |
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Definition
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Term
| Show that a piecewise function is differentiable at the point a where the function rule splits. |
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Definition
~ Derive each equation.
~ Plug in x to both.
~ Set equal to each other. |
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Term
| Given s(t) (position function), find v(t). |
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Definition
~ Derive.
1-Position
2-Velocity
3-Acceleration |
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Term
| Given v(t), find how far a particle travels on [a,b]. |
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Definition
~ Keep everything positive.
~ Antiderive (Integral). |
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Term
| Find the average velocity (average rate) of a particle on [a,b]. |
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Definition
s(b)-s(a)
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b-a |
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Term
| Given v(t), determine if a particle is speeding up at t=k. |
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Definition
~ Derivative of v(t) to get acceleration.
~ Plug k into both.
~ If both are positive it is speeding up.
~ If they are both positive and negative it is slowing down. |
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Term
| Given v(t) and s(0), find s(t). |
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Definition
~ Antiderive v(t) to get s(t).
~ Plug in s(0) to find C. |
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Term
| Show that Rolle's Theorem holds on [a,b]. |
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Definition
f'(c)=0
~ Mean Value Theorem.
~ Y values are the same f(a)=f(b).
~ Continuous and differentiable. |
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Term
| Show that Mean Value Theorem holds on [a,b]. |
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Definition
f'(c)=f(b)-f(a)
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b-a
~ Continuous and differentiable. |
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Term
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Definition
~ (-∞,∞)
~ All possible x's unless a domain issue. |
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Term
| Find range (y values) of f(x) on [a,b]. |
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Definition
| ~ Find relative/absolute max & mins. |
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Term
| Find range of f(x) on (-∞,∞). |
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Definition
| ~ Find relative/absolute max & mins. |
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Term
| Find f'(x) by definition. |
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Definition
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Term
| Find derivative of inverse to f(x) at x=a. |
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Definition
~ Switch x and y.
~ Find derivative. |
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Term
| Find the absolute maximum of f(x) on [a,b]. |
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Definition
| ~ Plug endpoints and critical values into original. |
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Term
Y is increasing proportionally to x.
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Definition
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Term
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Definition
| ~ 2nd Fundamental Theorem. |
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Term
| The rate of change of population is..... |
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Definition
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Term
| The line y=mx+b is tangent to f(x) at (a,b). |
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Definition
| ~ Find tangent line and see if its true. |
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Term
| Find area using left Reimann sums. |
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Definition
| AreaL= f(a)[image]x+f(x1)[image]x+f(x2)[image]x...... |
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Term
| Find area using right Reimann sums. |
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Definition
| AreaR= [image]xf(x1)+[image]xf(x2)+...... |
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Term
| Find area using midpoint rectangles. |
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Definition
~ Find midpoints.
~ Add them together.
~ Multiply by [image]x. |
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Term
| Find area using trapezoids. |
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Definition
| Δx/2[f(a) + f(x)]+ Δx/2[f(x) + f(x2)] + Δx/2[f(x2) + f(x3)]... |
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Term
| Solve the differential equation.... |
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Definition
~ Move the x to one side and y's to the other.
~ Antiderive. |
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Term
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Definition
| ~ Area underneath the curve from a to x. |
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Term
| Given a base, cross sections perpendicular to the x-axis are squares. |
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Definition
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Term
| Find where the tangent line to f(x) is horizontal. |
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Definition
~ Derivative.
~ Set equal to zero. |
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Term
| Find where the tangent line to f(x) is vertical. |
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Definition
~ Derive.
~ Set equal to 1/0 (underfined). |
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Term
| Find the minimum acceleration given v(t). |
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Definition
~ Find f''' to get critical values.
~ Plug endpoints and critical values into acceleration. |
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Term
| Approximate the value of f(0.1) by using the tangent line to f at x=0. |
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Definition
~ Find tangent line.
~ Plug in value (f(.1)). |
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Term
| Find the derivative of f(gx)). |
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Definition
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Term
| Given a picture of f'(x), find where f(x) is increasing. |
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Definition
| ~ When the line is above the x-axis. |
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Term
| Given v(t) and s(0), find the greatest distance from the origin of a particle on [a,b]. |
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Definition
~ Antiderive.
~ Plug in points and see which has the greatest value. |
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Term
| Given a chart of x and f(x) on selected values between a and b, estimate f'(c) where c is between a and b. |
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Definition
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Term
| Find the area between curves f(x), g(x) on [a,b]. |
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Definition
| ~ Antiderivative of top-bottom. |
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Term
| Find the volume if the area between f(x), g(x) is rotated about the x-axis. |
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Definition
| ~ Antiderivative of top2-bottom2 with ∏ in front. |
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