Question

1. let (g(x)=4x^{5}-2x^{4}+3x - 1). which of the following limit statements about the end - behavior of (g) is correct? (a) (lim_{x

ightarrow-infty}g(x)=-infty) and (lim_{x
ightarrowinfty}g(x)=-infty) (b) (lim_{x
ightarrow-infty}g(x)=-infty) and (lim_{x
ightarrowinfty}g(x)=infty) (c) (lim_{x
ightarrow-infty}g(x)=infty) and (lim_{x
ightarrowinfty}g(x)=-infty) (d) (lim_{x
ightarrow-infty}g(x)=infty) and (lim_{x
ightarrowinfty}g(x)=infty) 12. let (h(x)=-2x(x - 3)^{2}(x + 4)^{3}). which of the following limit statements about the end - behavior of (h) is correct? (a) (lim_{x
ightarrow-infty}h(x)=-infty) and (lim_{x
ightarrowinfty}h(x)=-infty) (b) (lim_{x
ightarrow-infty}h(x)=-infty) and (lim_{x
ightarrowinfty}h(x)=infty) (c) (lim_{x
ightarrow-infty}h(x)=infty) and (lim_{x
ightarrowinfty}h(x)=-infty) (d) (lim_{x
ightarrow-infty}h(x)=infty) and (lim_{x
ightarrowinfty}h(x)=infty) 13. let the polynomial (f) be an odd function such that (f(-4)=5) is the location of a local minimum. which of the following statements must be true? (a) (f(4)=-5) is the location of a local minimum. (b) (f(4)=-5) is the location of a local maximum. (c) (f(4)=5) is the location of a local minimum. (d) (f(4)=5) is the location of a local maximum. 14. let (h) be the piecewise defined function shown above. what is the average rate of change of (h) over the interval (0leq xleq5)? (h(x)=\begin{cases}3x^{2}+1, &\text{if }x < 4\\4x - 9, &\text{if }xgeq4end{cases}) (a) 2 (b) 4 (c) 6 (d) 15 worksheet d: topics 1.1 - 1.6 polynomial review created by bryan passwate

Explanation:

Step1: Analyze end - behavior of polynomial $g(x)$

For a polynomial $g(x)=4x^{5}-2x^{4}+3x - 1$, the leading term is $4x^{5}$. Since the degree $n = 5$ (odd) and the leading - coefficient $a=4>0$, as $x\to-\infty$, $g(x)\to-\infty$ and as $x\to+\infty$, $g(x)\to+\infty$.

Step2: Analyze end - behavior of polynomial $h(x)$

For the polynomial $h(x)=-2x(x - 3)^{2}(x + 4)^{3}$, expand it or consider the sign of the leading term. The degree of the polynomial is $1 + 2+3=6$ (even) and the leading - coefficient is negative (because of the $-2$ in front). So, as $x\to-\infty$, $h(x)\to-\infty$ and as $x\to+\infty$, $h(x)\to-\infty$.

Step3: Use properties of odd functions

If $y = f(x)$ is an odd function, then $f(-x)=-f(x)$. Given $f(-4)=5$ is a local minimum, then $f(4)=-f(-4)=-5$. Also, for an odd - function, the graph is symmetric about the origin. So, if $(-4,5)$ is a local minimum, then $(4, - 5)$ is a local maximum.

Step4: Calculate average rate of change of $h(x)$

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Here, $a = 0$, $b = 5$.
For $x = 0$, $h(0)=3(0)^{2}+1 = 1$ (since $0<4$).
For $x = 5$, $h(5)=4(5)-9=11$ (since $5\geq4$).
The average rate of change is $\frac{h(5)-h(0)}{5 - 0}=\frac{11 - 1}{5}=\frac{10}{5}=2$.

Answer:

1. B
2. A
3. B
4. A