IB Mathematics Applications & Interpretation Question Type 7: Formulating Piecewise Models Based on Contextual Information and Descriptions Exercises

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

Let $$f(x)=\{\begin{array}{ll}2x+a,& x\le 1,\\ x^2-3,& 1<x<4,\\ ax,& x\ge 4.\end{array}$$ Determine whether there exists a value of $a$ such that $f$ is continuous on $\mathbb{R}$.

Find the relationship between $a$ and $b$ such that the function $f(x)$ defined below is continuous at $x=1$. $$f(x)=\{\begin{array}{ll}{x}^2-ax+b,& x<1\\ ax-b,& x\ge 1\end{array}$$

Find the relationship between the constants $c$ and $d$ such that $$f(x)=\{\begin{array}{ll}{x}^2-2x+c,& x\le 2\\ 3x+d,& x>2\end{array}$$ is continuous on $\mathbb{R}$.

Find $a$ and $b$ so that $f$ is continuous on $\mathbb{R}$.

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.

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 $x>7$.

Determine the domain and range of the cost function $C(x)$.

Solve for all values of $x$ such that $C(x)=100$.

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

For what values of $k$ and $m$ is the function $$f(x)=\{\begin{array}{ll}1-x,& x<3\\ k(x-2{)}^2+m,& x\ge 3\end{array}$$ continuous at $x=3$?

Recall the piecewise cost function $C(x)$ defined as: $$C(x)=\{\begin{array}{ll}100{\left(\frac{1}{2}\right)}^x& \text{for }0\le x\le 4\\ 6.25& \text{for }4<x\le 7\\ 6.25+3(x-7)& \text{for }x>7\end{array}$$ Calculate $C(2)$, $C(5)$, and $C(10)$.