How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

oleifere45
2022-06-25
Answered

How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions?

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Paxton James

Answered 2022-06-26
Author has **25** answers

If $f$ is an arbitrary continuous funciton, then you can define a function $g$ via the integral and by the fundamental theorem, the derivative of $g$ is $f$. That is: $g$ is an antiderivative of $f$. That's all.

asked 2022-07-08

What is the difference between first and second fundamental theorem of calculus?

asked 2022-06-24

We know if $g$ is continuous on $(a,b)$ and $F(x)={\int}_{a}^{x}g(t)dt$, then

${F}^{\prime}(x)=g(x)$

But, how about if we have

$F(x)={\int}_{a}^{h(x)}g(t)dt$

What should ${F}^{\prime}(x)$ be?? can we still apply fundamental theorem of calculus?

${F}^{\prime}(x)=g(x)$

But, how about if we have

$F(x)={\int}_{a}^{h(x)}g(t)dt$

What should ${F}^{\prime}(x)$ be?? can we still apply fundamental theorem of calculus?

asked 2022-05-13

Calculate:

${\int}_{-2}^{2}\mathrm{sin}({x}^{5}){e}^{({x}^{8}\mathrm{sin}({x}^{4}))}dx$

${\int}_{-2}^{2}\mathrm{sin}({x}^{5}){e}^{({x}^{8}\mathrm{sin}({x}^{4}))}dx$

asked 2022-06-08

Suppose $F(x)={\int}_{3x+8}^{{x}^{2}+5x+1}{\mathrm{csc}}^{2}\left(t\right)dt$. How would one find ${F}^{\prime}(x)$ using the first fundamental theorem of calculus?

asked 2022-07-01

Let $f$ be continuous on $I=[a,b]$ and let $H:I\to \mathbb{R}$ be defined by $H(x)={\int}_{x}^{b}f(t)dt,x\in I$. Find ${H}^{\prime}(x)$.

asked 2022-05-10

Use the fundamental theorem of calculus:

Find ${F}^{\prime}(x)$, where $F(x)={\int}_{0}^{{x}^{3}}{\mathrm{sin}}^{3}t\phantom{\rule{thickmathspace}{0ex}}dt$

Find ${F}^{\prime}(x)$, where $F(x)={\int}_{0}^{{x}^{3}}{\mathrm{sin}}^{3}t\phantom{\rule{thickmathspace}{0ex}}dt$

asked 2022-05-10

Are there any cases which the First Fundamental Theorem of Calculus would fail?