(a)Express f in standard form

(b)Find the vertex and x- and y-intercepts of f.

(c)Sketch the graph.

(d)Find the domain and range of f.

preprekomW
2021-09-14
Answered

A quadratic function f is given

(a)Express f in standard form

(b)Find the vertex and x- and y-intercepts of f.

(c)Sketch the graph.

(d)Find the domain and range of f.

$f\left(x\right)={x}^{2}-2x+3$

(a)Express f in standard form

(b)Find the vertex and x- and y-intercepts of f.

(c)Sketch the graph.

(d)Find the domain and range of f.

You can still ask an expert for help

Yusuf Keller

Answered 2021-09-15
Author has **90** answers

(1)(a)

On comparing it with equation (1) we get:

So, we have (0,3) as the y−intercept.

Now the x−intercept is given by equating

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Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.

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Can this ODE system be solved?

$x}^{\prime}\left(t\right)=\mathrm{sin}(x\left(t\right)(\frac{y\left(t\right)}{2}+1);{y}^{\prime}\left(t\right)=\frac{-\mathrm{cos}\left(x\left(t\right)\right)\mathrm{cos}\left(y\left(t\right)\right)}{2$

Is there a method to solve the following ODE system?

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$\frac{du}{dt}-2tu=1$

My Attempt

I will verify this by differentiating the function with respect to t.

$\begin{array}{rl}\frac{du}{dt}& =\frac{d}{dt}({e}^{{t}^{2}}{\int}_{0}^{t}{e}^{-{s}^{2}}ds+{e}^{{t}^{2}})\\ & =2t{e}^{{t}^{2}}\left({\int}_{0}^{t}{e}^{-{s}^{2}}ds\right)+{e}^{{t}^{2}}\frac{d}{dt}\left({\int}_{0}^{t}{e}^{-{s}^{2}}ds\right)+2t{e}^{{t}^{2}}\end{array}$

I'm having trouble solving the integral because it involves an error function. Could I get some pointers on how to evaluate this? Or is there a different way to verify that the function is a solution?

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Showing that continuous and differentiable s(t), with $s\left(0\right)=0$ and $s}^{\prime}\left(t\right)\le 2ts\left(t\right)+\sqrt{s\left(t\right)$ for $t>0$, is identically zero for $t\ge 0$.

Let $s:[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ be a continuous function that is differentiable on $(0,+\mathrm{\infty})$.

It also holds that ${s}^{\text{'}}\left(t\right)\le 2ts\left(t\right)+\sqrt{s\left(t\right)},\text{}\text{}t0s\left(0\right)=0$

Show that $s\left(t\right)=0,t\ge 0$.

Do we us some theorem, maybe Rolle or the mean value theorem?

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Constant: $c>0$ such that for all $C}^{1$ function in $(0,1)$

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