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Class Notes Calculus I

Review Notes

Review Notes for Test.

Rate of temperature change proportional to temperature: dT/dt = k T. Rate of depth change proportional to square root of depth: dy/dt = k time square root of y.

 

Rate of population change proportional to product of population and difference between population and carrying capacity. dP/dt = k P ( L - P ).

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Tangent line at t sub 0 point: derivative at t sub 0 equal to ( y - f(t sub 0) ) / ( x - t sub 0).

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Approximate solution to dy/dt = f(y, t): Start at y sub 0, t sub 0, evaluate dy/dt, find delta y = dy/dt multlied by delta t; new y value equal old y value plus delta y. Repeat with t = t sub 0 plus delta t, and new y value. Iterate.

 

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Apply the difference quotient to find the derivative of y = x squared.

 

The units of dy/dx are units of y divided by units of x. dy/dx is the change in y per unit change in x.

 

The second derivative is approximated by the change in average slope divided by the change in midpoint value of t. A function which is concave up has positive second derivative; concave down implies negative second derivative.

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Difference between left- and right-hand sums is (f(b) - f(a) ) multiplied by delta t, and can be represented by a rectangle with these dimensions.

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Doubling time for y = A b^t is log 2 divided by log b. Doubling time for e^(k t) is natural log of 2 divided by k.

 

y = a t^2 + b t + c has zeros at t = (- b +- square root of (b^2 - 4 a c) ) divided by (2 a), with vertex on the vertical line t = -b / (2 a).

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y coordinate of motion on a circle of radius A centered at the origin, beginning at angular position theta sub 0, with period T has equation y = A sin(2 pi / T * t + theta sub 0).

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relationship between sine of t and inverse sine of t.

 

The polynomial y = c ( x - x sub 1) ( x - x sub 2) ( x - x sub 3) has zeros at x coordinates x sub 1, x sub 2 and x sub 3. The graph of the function crosses the x axis at these coordinates.

 

y = c ( x - x sub 1) / (x - x sub 2) has zero at x sub 1 and vertical asymptote as x = x sub 2.

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