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Linear Algebra

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.

(b) Eliminate the parameter to find a Cartesian equation of the curve

x=\sqrt{t}, \quad y=1-t

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Linear Algebra

In an election campaign, the popularity, B, as a percent of voters, of the governing Blue party can be modelled by a function of time, t, in days throughout the campaign as B(t) = 40 - 0.5t. The popularity, R(t), of the opposing Red party can be modelled by a composite function of B(t),R(B(t)) = 20 + 0.75[40 – B(t)].

Graph B(t) and describe the trend.

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Linear Algebra

\text { 4] Find } g(0)-g(9)+g(2) \text {, if }

g(x)=\left\{\begin{aligned}
\frac{x+1}{2}, & \text { if } x \text { is odd } \\
\frac{x}{2}, & \text { if } x \text { is even }
\end{aligned}\right.

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Linear Algebra

\text { 3] Find } f(4)-f(2)+f(3) \text {, if }

f(x)=\left\{\begin{array}{ll}
\frac{x+1}{2}, & \text { if } x \text { is odd } \\
\frac{x}{4}, & \text { if } x \text { is even }
\end{array}\right.

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Linear Algebra

2] Find the Domains of the following functions:

\text { a) } f(x)=\sqrt{15-5 x}

f(x)=\frac{x^{2}-2 x+1}{x^{2}-4 x-21}

f(x)=\frac{x^{2}-2 x+1}{\sqrt{16-2 x}}

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Linear Algebra

a) Give the definition of a rational function. [5 pts]

b) Give an example of a polynomial function of degree 3. [5 pts]

c) Can a constant function be a polynomial and a rational function at the same time? Explain your answer. [5 pts]

d) Give an example of a non-polynomial function and explain why not apolynomial function. [10 pts)

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Linear Algebra

5) Ве,

\begin{array}{c}
f: \mathbb{R} \rightarrow \mathbb{R} \\
f(x)=\left\{\begin{array}{ll}
x^{2}-3 \cos (\pi x) & x<0 \\
x-4 e^{-2 x} & x \geq 0 .
\end{array}\right.
\end{array}

Calculate

\int_{-1}^{2} f(x) d x

Presenting the result in simplified form

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Linear Algebra

4) Be f: R - Ra function differentiable in R such your derivative f', has in maximum a real zero.

Prove that the equation f(x)=0 has in maximum 2 real square

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Linear Algebra

3) Take in consideration the following function

f(x)=\left\{\begin{array}{ll}
f: \mathbb{R} \rightarrow \mathbb{R} & \\
\frac{x^{2}-4 x+\cos (\sin (x)),}{x^{4}+4 x^{2}+1}, & x \leq 0
\end{array}\right.

a) Show that the function f is continuous in R+ and in R- but discontinuous in the point X

b) Say justifying, if f is differenciable in X=0

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Linear Algebra

2) Prove by definition that

\lim _{x \rightarrow 0} x^{2} \cos \left(e^{x}\right)=0

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Linear Algebra

1) Calculate the following limit:

\lim _{x \rightarrow+\infty} \frac{x^{2}\left(e^{-3 x}+1\right)+x \cos (5 x)}{x^{2}+7 x+1}

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**Use of solution provided by us for unfair practice like cheating will result in action from our end which may include permanent termination of the defaulter’s account.Disclaimer:The website contains certain images which are not owned by the company/ website. Such images are used for indicative purposes only and is a third-party content. All credits go to its rightful owner including its copyright owner. It is also clarified that the use of any photograph on the website including the use of any photograph of any educational institute/ university is not intended to suggest any association, relationship, or sponsorship whatsoever between the company and the said educational institute/ university. Any such use is for representative purposes only and all intellectual property rights belong to the respective owners.

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