Calculus I Examples for Exam V

Calculus I Examples for Exam V

Here is a Maple Worksheet on differentiating and integrating logarithmic and exponential functions.

Consider f(x) = a^x.

f '(x) = a^xln(a)

f '(0) = a^oln(a) = ln(a)

Thus the derivative of an exponential function of the form f(x) = a^x will be one when x = 0 if a = e. Here is an applet that draws successive graphs of f(x) = a^x for increasing values of a starting at a = 1 along with the graph of y = x + 1. If you watch closely you can observe that for some value of a the graph of y = x + 1 is tangent to the graph of f(x) = a^x at (0,1). In this demonstration the number e is taken to be the value for a where the graphs of f(x) = a^x and y = x + 1 are tangent at (0,1).

Ln(x) animation In the animation y = ln(x) (in purple) and y = 1/x (in blue) are graphed in the same coordinate system. Coordinated blue animated points are moving along the graph of each. There is a vertical blue line segment extending from the point (1,0) to the point (1,1). There is an animated vertical line segment coordinated with the two animated points (i.e., same value for x). The area between the x-axis, the vertical blue line segment, the graph of y = 1/x, and the animated vertical line segment represents the absolute value of ln(x) where x is the x value along the animated line segment. The value of ln(x) will be negative where the animated line segment is red and positive where the animated line segment is green. I will go over this in class. Quicktime animation

Ln(x) animation2 In this version the region whose area corresponds to Ln(x) is shaded, red when Ln(x) is negative and green when Ln(x) is positive. Quuicktime version

Differentiation Examples

Integration Examples

Keep in mind that in integrating a fraction, if the derivative of the denominator is a constant times the numerator then you can use the natural log function (ln) in doing the integration.

The pictures below relate to the example directly above. The scales on the x-axis and y-axis are not the same so the pictures could be a little misleading but the areas represented would be the same.

Implicit Differentiation Example

Below is a graph of the relation.

Area Between Two Curves

Area Between Two Curves-Approximation

Maple Worksheet

Exponential Population Growth

The population of a country is growing at a rate that is proportional to the population of the country. The population in 1990 was 20 million and in 2000 the population was 22 million. Estimate the population in 2020.

Solution

Differential Equations Sailing Application Example

NEWTON'S LAW OF COOLING PROBLEM: A pot of liquid is put on the stove to boil. The temperature of the liquid reaches 170^oF and then the pot is taken off the burner and placed on a counter in the kitchen. The temperature of the air in the kitchen is 76^oF. After two minutes the temperature of the liquid in the pot is 123^oF. How long before the temperature of the liquid in the pot will be 84^oF? Click here for more on Newton's Law of Cooling. Solution Function Graph

Solving a Separable Differential Equation

The graph of the solution is shown above. Click on the graph to see an animation of the direction field vectors moving across the screen for increasing values of x along with an animated solution point. Solutions corresponding to initial conditions y(0) = -1, -0.4, 0.4, 1 are shown below along with the direction field. Click on the picture to see an enlargement.

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu