91Ó°ÊÓ

(a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given \(x\) -coordinate and \((\boldsymbol{b})\) find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. HINT [See Example \(2(\mathrm{~b}) .]\) $$ f(x)=\frac{1}{x^{2}} ; x=1 $$

If the average rate of change of a function over \([a, b]\) is zero, this means that the function has not changed over \([a, b]\), right?

Sketch the graph of a function whose average rate of change over \([0,3]\) is negative but whose average rate of change over \([1,3]\) is positive.

Of the three methods (numerical, graphical, algebraic) we can use to estimate the derivative of a function at a given value of \(x\), which is always the most accurate? Explain.

Explain why we cannot put \(h=0\) in the formula $$ f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ for the derivative of \(f\).

The price per barrel of crude oil in constant 2008 dollars can be approximated by $$ P(t)=0.45 t^{2}-12 t+105 \text { dollars } \quad(0 \leq t \leq 28) $$ where \(t\) is time in years since the start of \(1980.5^{5}\) a. Compute the average rate of change of \(P(t)\) over the interval \([0,28]\), and interpret your answer. HINT [See Section \(10.4\) Example 3.] b. Estimate the instantaneous rate of change of \(P(t)\) at \(t=0\), and interpret your answer. HINT [See Example 2(a).] c. The answers to part (a) and part (b) have opposite signs. What does this indicate about the price of oil?

Company A's profits are given by \(P(0)=\$ 1\) million and \(P^{\prime}(0)=-\$ 1\) million/month. Company B's profits are given by \(P(0)=-\$ 1\) million and \(P^{\prime}(0)=\$ 1\) million/month. In which company would you rather invest? Why?

Use the difference quotient to explain the fact that if \(f\) is a linear function, then the average rate of change over any interval equals the instantaneous rate of change at any point.

Sketch the graph of a function whose derivative is never negative but is zero at exactly two points.

Here is the graph of the derivative \(f^{\prime}\) of a function \(f\). Give a rough sketch of the graph of \(f\), given that \(f(0)=0\).