For the following exercise, perform the operations. Express answers in reduced form.

$$\dfrac{1}{2 x}-\dfrac{2}{3}$$

Can the formula in the following exercise be written in the form of an exponential function or a power function? If not, explain why the function does not fit either form.

$$q(x)=5^{\left(x^2\right)}$$

Describe the behavior of y=x$^{-10}$ and y=-x$^{10}$ as

(a) $x \rightarrow 0$

(b) $x \rightarrow \infty$

(c) $\quad x \rightarrow-\infty$

What is the long-run behavior of the functions in the following problem?

$$y=\frac{x(x+5)(x-7)}{4+x^2}$$

In this problem, give a possible formula for the polynomial.

(-2,0), (-1,-3), (2,0), (3,0)

Anthropologists suggest that the relationship between the body weight and brain weight of primates can be modeled with a power function. The table below lists various body weights and the corresponding brain weights of different primates.

(a) Using the table below, find a power function that gives the brain weight, $Q$ (in $\mathrm{mg}$ ), as a function of the body weight, $b$ (in gm).

(b) The erythrocebus (Patas monkey) has a body weight of $7800 \mathrm{gm}$. Estimate its brain weight.

In the following exercise, does the function represent proportionality to a power of the independent variable? That is, can the function be written in the form $y=k x^p$ for the variables given in the problem? If so, identify the constant, $k$, and the power, $p$.

$$z=5(3)^x$$

Are the function in the following exercise power functions? If so, write the function in the form $f(x)=k x^p$.

$$Q(t)=\left(\frac{1}{2 \sqrt{t}}\right)^3$$

Is the function in the following exercise polynomial? If so, of what degree?

$$y=4 x^2+2$$

Write $y=3\left(\frac{2}{5 \sqrt{7 x}}\right)^4$ in the form $y=a x^p$ and state the values of a and p.

A function that is not a polynomial can often be approximated by a polynomial. For example, for certain x values, the function $f(x)=e^x$ can be approximated by the fifth-degree polynomial

$$p(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120} .$$

(a) Show that $p(1) \approx f(1)=e$. How good is the estimate?

(b) Calculate $p(5)$. How well does $p(5)$ approximate $f(5) ?$

(c) Graph $p(x)$ and $f(x)$ together on the same set of axes. Based on your graph, for what range of values of $x$ does $p(x)$ give a good estimate for $f(x)$ ?

The following problem give value of transformations of either y= 1 / x or y=1 / x$^2$. In each case

(a) Determine if the values are from a transformation of $y=1 / x$ or $y=1 / x^2$. Explain your reasoning.

(b) Find a possible formula for the function.

(a) Find the average rate of change of f(x)=x$^{1 / 2}$ between successive values of x for x=0,2,4,6,8. Do the same for g(x)=x$^2$.

(b) What do these rates of change suggest about the concavity of the graphs of f(x)=x$^{1 / 2}$ and g(x)=x$^2$ ?

In this problem, give a possible formula for the polynomial.

(-1,0), (0,0),(2,0)