find the antiderivative for each function when c equals 0. a. f(x)=e^4x b. g(x)=e^ - 10x c. h(x)=e^x/9

Explanation:

Step1: Recall antiderivative formula

The antiderivative of $e^{ax}$ is $\frac{1}{a}e^{ax}+C$. Here $C = 0$.

Step2: Find antiderivative of $f(x)$

For $f(x)=e^{4x}$, using the formula with $a = 4$, we have $\int e^{4x}dx=\frac{1}{4}e^{4x}$.

Step3: Find antiderivative of $g(x)$

For $g(x)=e^{- 10x}$, with $a=-10$, $\int e^{-10x}dx=-\frac{1}{10}e^{-10x}$.

Step4: Find antiderivative of $h(x)$

For $h(x)=e^{\frac{x}{9}}$, where $a=\frac{1}{9}$, $\int e^{\frac{x}{9}}dx = 9e^{\frac{x}{9}}$.

Answer:

a. $\frac{1}{4}e^{4x}$ b. $-\frac{1}{10}e^{-10x}$ c. $9e^{\frac{x}{9}}$