Answer
\begin{aligned}
& \text{Question: If } f(1) = 12, f' \text{ is continuous, and } \int_{1}^{4} f'(x)dx = 17, \text{ what is the value of } f(4)? \\
& \text{Answer: } f(4) = 29
\end{aligned}
Work Step by Step
\begin{aligned}
& \text{According to the Fundamental Theorem of Calculus, if } f \text{ is continuous on } [a,b] \text{ and } F \text{ is an antiderivative of } f \text{ on } [a,b], \\
& \text{then } \int_{a}^{b} f(x)dx = F(b) - F(a). \\
& \text{In this case, we know that } f' \text{ is continuous and } \int_{1}^{4} f'(x)dx = 17. \\
& \text{So, if we let } F(x) \text{ be an antiderivative of } f'(x), \\
& \text{then we have } F(4) - F(1) = 17. \\
& \text{Since } F(x) \text{ is an antiderivative of } f'(x), \\
& \text{we know that } F(x) = f(x) + C \text{ for some constant } C. \\
& \text{So, we can rewrite the above equation as } f(4) + C - (f(1) + C) = 17. \\
& \text{Solving for } f(4) \text{ and using the fact that } f(1) = 12, \\
& \text{we get:} \\
f(4) + C - (f(1) + C) &= 17 \\
f(4) - f(1) &= 17 \\
f(4) - 12 &= 17 \\
f(4) &= 29
\end{aligned}
