Use Newton’s method to find the positive fourth root of 2 by solving the equation $x^{4}-2=0.$ Start with $x_{0}=1$ and find $x_{2}.$

Determine all critical points for each function. $y=x^{2}-32 \sqrt{x}$

It took 14 sec for a mercury thermometer to rise from $-19^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of $8.5^{\circ} \mathrm{C} / \mathrm{sec}.$

Graph the function over the given domain. Which of the extreme values, if any, are absolute?

$$f(t)=12 t-t^3, \quad-3 \leq t<\infty$$

What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have?

You are planning to make an open rectangular box from an 24-cm-$ by-45-cm piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of the largest volume you can make this way, and what is its volume?

Identify the function’s local extreme values in the given domain, and say where they occur.

$$k(x)=x^{3}+3 x^{2}+3 x+1, -\infty<x \leq 0$$

a. Identify the function’s local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $f(t)=t^{3}-3 t^{2}, \quad-\infty<t \leq 3$

When metallic tin is kept below $13.2^{\circ} \mathrm{C},$ it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious, and indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases, it is reasonable to assume that the rate y = dx/dt of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, y may be considered to be a function of x alone, and $v=k x(a-x)=k a x-k x^{2},$ where x = the amount of product a = the amount of substance at the beginning k = a positive constant. At what value of x does the rate y have a maximum? What is the maximum value of y?

You are planning to make an open rectangular box from an 8-in.-by-15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume?

Answer for the following function whose derivative is given:

At what points, if any, does $f$ assume local maximum or minimum values?

$$f^{\prime}(x)=(x-1)^2(x+2)^2$$

Answer the following question about the functions whose derivatives are given in the following exercise:

a. What are the critical points of $f$ ?

b. On what open intervals is $f$ increasing or decreasing?

c. At what points, if any, does $f$ assume local maximum and minimum values?

$$f^{\prime}(x)=(x-1)(x+2)(x-3)$$

Which of the extreme values, if any, are absolute?

$$k(x)=x^{3}+3 x^{2}+3 x+1, -\infty<x \leq 0$$

Identify the function's local extreme values in the given domain, and say where they occur.

$$f(t)=12 t-t^3, \quad-3 \leq t<\infty$$