91Ó°ÊÓ

Explain why the Mean Value Theorem does not apply to the function \(f\) on the interval \([0,6]\). $$f(x)=|x-3|$$

Mean Value Theorem Consider the graph of the function \(f(x)=x^{2}+1 .\) (a) Find the equation of the secant line joining the points \((-1,2)\) and \((2,5) .\) (b) Use the Mean Value Theorem to determine a point \(c\) in the interval \((-1,2)\) such that the tangent line at \(c\) is parallel to the secant line. (c) Find the equation of the tangent line through \(c .\) (d) Then use a graphing utility to graph \(f\), the secant line, and the tangent line.

Mean Value Theorem Consider the graph of the function \(f(x)=-x^{2}-x+6 .\) (a) Find the equation of the secant line joining the points \((-2,4)\) and \((2,0) .\) (b) Use the Mean Value Theorem to determine a point \(c\) in the interval \((-2,2)\) such that the tangent line at \(c\) is parallel to the secant line. (c) Find the equation of the tangent line through \(c\). (d) Then use a graphing utility to graph \(f\), the secant line, and the tangent line.

A shampoo bottle is a right circular cylinder. Because the surface area of the bottle does not change when it is squeezed, is it true that the volume remains the same? Explain.

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Find the limit. (Hint: Let \(x=1 / t\) and find the limit as \(t \rightarrow 0^{+}\).) $$ \lim _{x \rightarrow \infty} x \sin \frac{1}{x} $$

Consider the function on the interval \((0,2 \pi)\). For each function, (a) find the open interval(s) on which the function is increasing or decreasing, (b) apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$f(x)=\frac{x}{2}+\cos x$$

The sum of the perimeters of an equilateral triangle and a square is \(10 .\) Find the dimensions of the triangle and the square that produce a minimum total area.

Find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. $$ \lim _{x \rightarrow-\infty}\left(x+\sqrt{x^{2}+3}\right) $$

Find the limit. (Hint: Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify your result. $$ \lim _{x \rightarrow-\infty}\left(3 x+\sqrt{9 x^{2}-x}\right) $$