Question Type 7: Formulating piecewise models based on contextual information and descriptions Exercises

Let $f(x)=\begin{cases}x^2 - ax + 3,&0< x<2,\\a - x,&x\ge2.\end{cases}$ For what value of $a$ is $f$ continuous on $[0,\infty)$?

Determine $a$ and $b$ such that

Determine the domain and range of the piecewise cost function $C(x)$ from Question 1.

Find $c$ and $d$ so that

Using the piecewise cost function $C(x)$ from Question 1, calculate $C(2)$, $C(5)$, and $C(10)$.

Write a piecewise function $C(x)$ for the cost of producing $x$ units, where the cost is initially 100 at $x=0$, decreases exponentially by a factor of $1/2$ for each unit produced for $0\\le x\le 4$, remains constant for $4< x\le 7$, and increases at a rate of 3 per unit for $x>7$.

Let

Find $a$ so that $f$ is continuous on $\mathbb R$.

Sketch the graph of the cost function $C(x)$ from Question 1, labeling the points at $x=0$, $x=4$, and $x=7$.

For what values of $k$ and $m$ is the function

Find the average rate of change of the cost function $C(x)$ from $x=2$ to $x=6$ using the piecewise model from Question 1.

Let

Solve for all $x$ such that $C(x)=100$ in the cost model from Question 1.