3.4: The Derivative Function
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This optional summary video from the textbook author might be helpful to use as a preview. Other, more detailed, supplemental videos for this section are posted at the end of the text.
Section 3.4
Supplemental videos
Homework Exercises 3.4
WeBWorK Problems:
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Written Problems:
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2. Let g(x) be a continuous function (that is, one with no jumps or holes in the graph) and suppose that a graph of y=g′(x) is given by the graph on the right in the figure below. Please sketch the following on your paper and label them as g and g'
a. Observe that for every value of x such that 0<x<2, the value of g′(x) is constant. What does this tell you about the behavior of the graph of y=g(x) on this interval?
b. On what intervals other than 0<x<2 do you expect y=g(x) to be a linear function? Why?
c. At which values of x is g′(x) not defined? What behavior does this lead you to expect to see in the graph of y=g(x)?
d. Suppose that g(0)=1. Sketch an accurate graph of y=g(x).
3. Consider the function g(x)=x^2-x+3
a. Use the limit definition of the derivative to determine a formula for g′(x).
b. Use a graphing utility to plot both y=g(x) and your result for y=g′(x). Sketch these on your paper. Annotate your graphs to show that the graphs makes sense in terms of slopes.
4. Let f be a function with the following properties:
- f is differentiable at every value of x (that is, f has a derivative at every point)
- f(−2)=1
- f′(−2)=−2
- f′(−1)=−1
- f′(0)=0
- f′(1)=1
- f′(2)=2.
Sketch a possible graph of y=f(x). Explain why your graph meets the stated criteria.