calculus bc mcq practice test b. on the graph of y = f(x), the slope at any point (x, y) is twice the value…

Explanation:

Step1: Write the derivative equation

The slope of $y = f(x)$ at any point $(x,y)$ is $f'(x)$. Given that $f'(x)=2x$.

Step2: Integrate to find $f(x)$

Integrate $f'(x)$ with respect to $x$. $\int f'(x)dx=\int 2x dx$. Using the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n = 1$ here), we get $f(x)=x^{2}+C$.

Step3: Find the value of $C$

Since $f(2) = 3$, substitute $x = 2$ and $f(2)=3$ into $f(x)=x^{2}+C$. So $3=2^{2}+C$, which gives $3 = 4+C$, and $C=-1$. Then $f(x)=x^{2}-1$.

Step4: Find $f(3)$

Substitute $x = 3$ into $f(x)=x^{2}-1$. So $f(3)=3^{2}-1=9 - 1=8$.

Answer:

C. 8