91Ó°ÊÓ

The function \(f(x)=\frac{x^{3}-1}{x-1}\) is graphed in Figure \(12.39 .\) It appears to be continuous on the interval \([-3,3],\) but there is an \(x\) -value on that interval at which the discontinuous. Determine the value of \(x\) at which the function is discontinuous, and explain the pitfall of utilizing continuity of a function by examining its graph.

Find the limit \(\lim _{x \rightarrow 1} f(x)\) and determine if the following function is continuous at \(x=1 :\) $$ f x=\left\\{\begin{array}{ll}{x^{2}+4} & {x \neq 1} \\ {2} & {x=1}\end{array}\right. $$

How is the slope of a linear function similar to the derivative?

What is the difference between the average rate of change of a function on the interval \([x, x+h]\) and the derivative of the function at \(x ?\)

A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?

Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.

For the following exercises, use the definition of derivative \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to calculate the derivative of each function. $$f(x)=3 x-4$$

For the following exercises, use the definition of derivative \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to calculate the derivative of each function. $$f(x)=-2 x+1$$

For the following exercises, use the definition of derivative \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to calculate the derivative of each function. $$f(x)=x^{2}-2 x+1$$

For the following exercises, use the definition of derivative \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) to calculate the derivative of each function. $$f(x)=2 x^{2}+x-3$$