PLEASE DO NUMBER THREE

Directions: Use Python to solve each problem, unless the question states otherwise. $($Template link$)$

Given the vector $($:$3+si{n}^3(t),3+co{s}^3(t)$:$)$ :

$($a$)$ Find the equation for the tangent line at the point where $t=\frac{}{4}.$

$($b$)$ Find the points on the graph where the tangent line has a slope of $1.$

$($c$)$ Sketch the graph of the vector function on tin$[0,8]$ and all tangent lines found in parts $($a$)$ and $($b$).$

Consider the curve given by $x^2+2x+3y^2=8$ :

$($a$)$ Find $\frac{dy}{dx}.$

$($b$)$ Find the points $(x$ and $y$ coordinates$)$ where the graph of the equation has a horizontal tangent line.

$($c$)$ Find the points $(x$ and $y$ coordinates$)$ where the graph of the equation has a vertical tangent line.

$($d$)$ Plot the curve $($using plot$\_$implicit$)$ and the tangent lines from parts $($b$)$ and $($c$).$

A bank account that accrues continuously compounded interest grows in value at a rate proportional to its current value. The current value is modeled by the exponential equation $y=y_0{e}^{rt},$ where $r$ is the annual interest rate, $t$ is time in years, and $y_0$ is the initial value of the account. Suppose after $2$ years, the account held $$10,000$ and after $5$ years, the account held $$12,000.$

$($a$)$ Solve a system of equations to find $r$ and $y_0$ for $y=y_0{e}^{rt}.($Note: it will help to assume $r$ is real in the symbols command.$)$

$($b$)$ Use this to determine when the account will reach a value of $$15,000.$

$($c$)$ Suppose $$10,000$ was the "inital" value and $$12,000$ was the value after $3$ years. Find $r$ and the value of the account $2$ years BEFORE the "initial" time. In a print statement, explain what you notice when comparing these answers to part $($a$).$

PLEASE DO NUMBER THREE