g2esebyy7

2022-05-01

Find the antiderivative for a function

Hi! I'm a little confused about a calculus exercise. Can someone please check if my logic is right?

The task: Find the antiderivative for a function f if f is a continuous function and if ${F}^{\prime}(x)=f(x)$ for all real numbers x, of ${\int}_{1}^{4}f(2x)\phantom{\rule{thinmathspace}{0ex}}dx$.

As far as I understand the answer is going to be $2F(4)-2F(1)$?

Hi! I'm a little confused about a calculus exercise. Can someone please check if my logic is right?

The task: Find the antiderivative for a function f if f is a continuous function and if ${F}^{\prime}(x)=f(x)$ for all real numbers x, of ${\int}_{1}^{4}f(2x)\phantom{\rule{thinmathspace}{0ex}}dx$.

As far as I understand the answer is going to be $2F(4)-2F(1)$?

Derick Alvarado

Beginner2022-05-02Added 10 answers

Step 1

There is no "the" antiderivative of a function f. If F is an antiderivative of f, then so is $F+C$ for any constant C.

The fundamental theorem of calculus tells you that if F is an antiderivative of f, then ${\int}_{p}^{q}f(x)dx=F(q)-F(p)$

Step 2

But now you have ${\int}_{a}^{b}f(2x)dx$. By a change of variable $\begin{array}{}\text{(1)}& {\int}_{a}^{b}f(2x)dx=\frac{1}{2}{\int}_{2a}^{2b}f(u)du\end{array}$

You can now apply the fundamental theorem of calculus to the right-hand side of (1).

For your problem, $a=1$, $b=4$.

There is no "the" antiderivative of a function f. If F is an antiderivative of f, then so is $F+C$ for any constant C.

The fundamental theorem of calculus tells you that if F is an antiderivative of f, then ${\int}_{p}^{q}f(x)dx=F(q)-F(p)$

Step 2

But now you have ${\int}_{a}^{b}f(2x)dx$. By a change of variable $\begin{array}{}\text{(1)}& {\int}_{a}^{b}f(2x)dx=\frac{1}{2}{\int}_{2a}^{2b}f(u)du\end{array}$

You can now apply the fundamental theorem of calculus to the right-hand side of (1).

For your problem, $a=1$, $b=4$.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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