5. Graphical Applications of Derivatives

Let ƒ(x) = (x - 3) (x + 3)²

f. State the x- and y-intercepts of the graph of ƒ.

Mean Value Theorem and graphs Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of f on [1,3]. Reconcile your results with the Mean Value Theorem. <IMAGE>

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

a. Graph and interpret the mileage function.

Which of the following statements best describes the behavior of a function $f$ that satisfies $f\left(0\right)=0$, $f^{\prime }\left(0\right)=3$, $f^{\prime }\left(1\right)=0$, and $f^{\prime }\left(2\right)=-1$?