Presentation exercises are marked with a *.²
(1) Exercises on limits at plus and minus infinity, from page 91 of the text, Exercises 1,2,3, 6*, 7, 11, 16, 17, 26, 27.
For exercises 6 and 7, use Mathematica to graph the two functions together in the same "window" by using the following code:
Needs["Graphics`Colors`"];
f[x_] := whatever
g[x_] := whatever
Plot[{f[x], g[x]}, {x, -7, 7}, PlotStyle-> {Red, Blue}]
Plot[{f[x], g[x]}, {x, -50, 50}, PlotStyle-> {Red, Blue}]
(2) Section 4.1, Exercises 1, 2, 3, 4, 5, 6, 9, 11, 12*, 13.
For exercises 9, 11, 12, 13, use the first derivative to find the critical points and the intervals on which the function is increasing and decreasing, and the second derivative to find the intervals on which the function is concave up and concave down. Using this information, make your own sketch of the graph of the function. Then use Mathematica to check your plot. Use the following code, where a and b are appropriate end points for the graph which will insure that the graph displayed has all the essential features displayed.
f[x_] := whatever
Plot[f[x], {x, a, b}]
(3) Section 4.2, Exercises 1,2 ,3 ,4, 14.
(4) Aditional Graphing exercises: for each of the following functions, compute the first and second derivatives. Find out on what intervals the function is increasing/decreasing and on what intervals the function is concave up/ concave down. Calculate the local maxima and minima, and the points of inflection (points where the concavity changes). Consider what happens the the function for large positive and negative values of x. On the basis of all this information, draw a sketch of the function, with the several important points labelled -- i.e. the local maxima and minima, and the points of inflection. Finally, get Mathematica to make a plot of the function and compare the computer plot with your sketch.
a) f(x) = 5 x^4 - 12 x^2 + 2x - 3. (Presentation exercise!) The problem here is that the first derivative is a cubic polynomial, so not so easy to solve f'[x] =0 by hand! However, you can get Mathematica to find the points at which f'[x] = 0 using the following code:
f[x_] := 5x^4 - 12x^2 + 2x
f'[x]
NSolve[f'[x] == 0, x]
b) f(x) = (7x -2)/(3x -10)