Calculus II Class 01/15


Calculus I Quiz 0115 

Find the general solution of dy/dt = 1.2 t, then give the specific solution for which y(0) = 10.

If dy/dt = 1.2 t then y is an antiderivative of 1.2 t, with respect to t.  The general antiderivative is

If y(0) = 10 we have

and our solution becomes

Find the general solution of dy/dt = 12 cos(t), then give the specific solution for which y(0) = 7.

Again we see that y is a general antiderivative with respect to t of 12 cos(t) so that

Since y(0) = 7 we have

and our particular solution is

Find the general solution of dy/dt = 1.2 / t, then give the specific solution for which y(1) = 2.

Our general solution is

Since y(0) = 2 we have

giving us particular solution

Find the definite integral of x^2 with respect to x, between x = 1 and x = t.  What is the derivative with respect to t of this result?

The definite integral of f(x) = x^2 with respect to x is the change in an antiderivative function between the two limits x = 1 and x = t.

Using antiderivative function F(x) = x^3 / 3 we find that the definite integral is

The derivative of this definite integral with respect to t is

Note that this is just f(t). 

This is an instance of the Second Fundamental Theorem of Calculus, shown below in standard notation and expressed in 'typewriter notation' as

In terms of a depth-and-rate interpretation, f(x) is the rate-of-depth-change function and the integral of this function between x = 1 and x = t is equal to the change in the depth function between x = 1 and x = t.

Find the definite integral of e^(2x) with respect to x, between x = 1 and x = t^2.  What is the derivative of this result?

Antiderivative is F(x) = 1/2 * e^(2x), so the definite integral is

The derivative of this result is d/dt [ 1/2 * e^(2 t^2) - 1/2 e^(2) ] = 2 t e^(2 t^2).

Had we followed the form of the last problem we might have first expected that the derivative would be just f(t), though on a little thinking we would see that t^2 is the upper limit, not t, so we might have expected f(t^2).  

The statement d/dt [ int(f(x), x, c, t ] = f(t) is a short statement of the Second Fundamental Theorem of Calculus, which we verified in the example before last for c = 1 and f(x) = x^2.

Represent the integral of e^(2x) with respect to x, from x = 0 to x = .5, as the area beneath a curve. 

The integral is represented as the area beneath the y = e^(2x) graph between x = 0 and x = .5, as in the figure below.

Represent in a similar manner the integral of the same function between x = 0 and x = .51. 

The integral is represented as the area beneath the y = e^(2x) graph between x = 0 and x = .51, as in the figure below.

The difference is represented as the area beneath the y = e^(2x) graph between x = .50 and x = .51, as in the figure below.

The average rate at which the integral changes with respect to change in the upper limit is

so in this case ave rate is represented by the area of the 'sliver' between x = .50 and x = .51, divided by the difference .01 between the upper limits x = .50 and x = .51.  This is depicted in the figure below.

The average rate is thus

That is, the average rate at which the integral of e^(2x) changes, with respect to the variable of integration, is equal to the value of e^(2x).