3. (3 pts) Given $$ m(2)=20, m^{\prime}(2)=-\frac{1}{3}, n^{\prime}(2)=-0.25, n(2)=-9 $$, and $$ g(x)=m(x) \cdot n(x) $$ a. Evaluate $$ g(2) $$. Show your work. b. Evaluate $$ g^{\prime}(2) $$. Show your work. Note: $$ g^{\prime}(2)=0.083 $$ is NOT correct.

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UpStudy AI · Accepted Answer

**(a) Evaluate $$ g(2) $$:**

Given:
- $$ m(2)=20 $$
- $$ n(2)=-9 $$

Since $$ g(x)=m(x) \cdot n(x) $$, we find:
$$
g(2)=m(2)\cdot n(2)=20\cdot (-9)=-180
$$

---

**(b) Evaluate $$ g^{\prime}(2) $$:**

Using the product rule where:
$$
g(x)=m(x)\cdot n(x) \quad \Rightarrow \quad g^{\prime}(x)=m^{\prime}(x)n(x)+m(x)n^{\prime}(x)
$$

Given:
- $$ m^{\prime}(2)=-\frac{1}{3} $$
- $$ n(2)=-9 $$
- $$ m(2)=20 $$
- $$ n^{\prime}(2)=-0.25 $$

Plug in the values:
$$
g^{\prime}(2)=\left(-\frac{1}{3}\right)(-9)+20(-0.25)
$$

Calculate each term:
$$
\left(-\frac{1}{3}\right)(-9)=3 \quad \text{(since } -\frac{1}{3}\times -9 = 3\text{)}
$$
$$
20(-0.25)=-5
$$

Thus:
$$
g^{\prime}(2)=3-5=-2
$$
**(a) Evaluate $$ g(2) $$:** $$ g(2) = m(2) \times n(2) = 20 \times (-9) = -180 $$ **(b) Evaluate $$ g^{\prime}(2) $$:** $$ g^{\prime}(2) = m^{\prime}(2) \times n(2) + m(2) \times n^{\prime}(2) = \left(-\frac{1}{3}\right) \times (-9) + 20 \times (-0.25) = 3 - 5 = -2 $$

(3 pts) Given ,
and
a. Evaluate . Show your work.
b. Evaluate . Show your work.
Note: is NOT correct.

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(3 pts) Given , and a. Evaluate . Show your work. b. Evaluate . Show your work. Note: is NOT correct.