Question

mata30h3f (lec 01 & 02) - final exam - practice 2

1. 8 points let $f(x) = 5 + 2x + \int_{-1}^{x^3} t^2 e^t dt$.

(a) determine all intervals where $f(x)$ is increasing.
(b) is $f(x)$ invertible?
(c) determine $f(-1)$.
(d) determine the value of $(f^{-1})(3)$.
a)

Explanation:

Step1: Compute f'(x) using FTC and chain rule

$f'(x) = 2 + (x^3)^2 e^{x^3} \cdot 3x^2 = 2 + 3x^8 e^{x^3}$

Step2: Analyze sign of f'(x)

$x^8 \geq 0$, $e^{x^3} > 0$, so $3x^8 e^{x^3} \geq 0$. Thus $f'(x) = 2 + \text{non-negative} > 0$ for all x.

Step3: Find f(-1) by substituting x=-1

$f(-1) = 5 + 2(-1) + \int_{-1}^{-1} t^2 e^t dt = 5 - 2 + 0 = 3$

Step4: Use inverse derivative formula

Since $f(-1)=3$, $(f^{-1})'(3) = \frac{1}{f'(-1)}$. Compute $f'(-1)=2 + 3(-1)^8 e^{(-1)^3}=2 + 3e^{-1}=2+\frac{3}{e}$

Answer:

(a) All real numbers, i.e., $(-\infty, \infty)$
(b) Yes
(c) 3
(d) $\frac{1}{2 + \frac{3}{e}}$ (or $\frac{e}{2e + 3}$)