the questions in level 1 are introductory problems. the hints contain links to videos covering this content.
question 1 (1 point)
suppose ( f(x)=8x^{3}+12x + 2 ) and ( f(1)=-4 ). then ( f(-1) ) equals
. (enter a number for your answer.)
view hint for question 1

Question

Accepted Answer

# Explanation:
## Step1: Integrate $f'(x)$
We know that if $f'(x)=8x^{3}+12x + 2$, then by the power rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), we have:
$$
\begin{align*}
f(x)&=\int(8x^{3}+12x + 2)dx\
&=8\int x^{3}dx+12\int xdx+2\int dx\
&=8\times\frac{x^{4}}{4}+12\times\frac{x^{2}}{2}+2x + C\
&=2x^{4}+6x^{2}+2x + C
\end{align*}
$$
## Step2: Find the value of $C$
Since $f(1)=-4$, substitute $x = 1$ into $f(x)=2x^{4}+6x^{2}+2x + C$:
$$
\begin{align*}
f(1)&=2\times(1)^{4}+6\times(1)^{2}+2\times(1)+C\
-4&=2 + 6+2+C\
-4&=10 + C\
C&=-14
\end{align*}
$$
So $f(x)=2x^{4}+6x^{2}+2x-14$
## Step3: Calculate $f(-1)$
Substitute $x=-1$ into $f(x)=2x^{4}+6x^{2}+2x-14$:
$$
\begin{align*}
f(-1)&=2\times(-1)^{4}+6\times(-1)^{2}+2\times(-1)-14\
&=2\times1+6\times1-2-14\
&=2 + 6-2-14\
&=-8
\end{align*}
$$

# Answer:
$-8$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Integrate \(f'(x)\)

Step2: Find the value of \(C\)

Step3: Calculate \(f(-1)\)

Answer: