Changing the value of "a" with the slider changes the base on the blue curve, which is y=c*a^(x-h) + k. The red curve is y=c*(1/a)^(x-h) + k, i.e. its base is the reciprocal of the base for the blue exponential function. The parameter "c" stretches/compresses the graphs vertically, while "h" slides the graph left/right and "k" moves them up and down. The beginning values of c, h, and k are set so that there is no stretching or sliding of graphs. The dotted green line (which is along the y=axis initially) is the line y=k, which is the horizontal asymptote of the graphs.

Use the slider to change the base--it goes from 0.1 up to 10, in increments of 0.1. What happens when a=1? How do the graphs' positions change when the based is changed from 2 to 0.5? Why? What happens as you increase c? What if c is negative? Can you move the curves up 3 units? Left 2 units?

Graphic for Example 1 Section 2.1. The graph is f(t) = 16t^2, the distance from starting position of an object in free-fall near the surface of the Earth.

Dotted segment slopes are average rates of change on [0,2] (pink), [0,1] (brown), and [1,2] (green), respectively. The point D (red) is movable, and the slope of the red line connecting D and (1,16) is the average rate of change in the time interval between 1 second and the time (first coordinate) at point D. As the point D moves closer to (1,16) from either the left or right, what happens to the slope of the red line?

This applet initially shows the function f(x)=x^2, and the area under the curve above the interval [a,b]. It starts with a=0 and b=2; the value of c is the shaded area, and the point (b,c) (a point on the antiderivative produced in the First Fundamental Theorem of Calculus) is plotted.

The purpose of this applet is to plot the antiderivative from FTC-1, even if you don't know a formula for it. You set the left and right limits on the plot in the obvious boxes; they are set initially to [-5,5]. Set "a" to be your lower limit of integration. The slider then moves "b" in increments of 0.1 units from the left limit of your plot to the right limit, tracing out the point (b,c) in red as it goes. These points (b,c) are on the graph y = F(x) from FTC-1, an antiderivative for f(x). Looking at the shape of the plot of red dots, you can enter a candidate formula for the antiderivative in the indicated box (initially this is set to the constant function y=0). This produces a dotted green plot that, if you are correct, will pass through all of the red dots.