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Exercises for Question Type 7: Formulating piecewise models based on contextual information and descriptions - IB | RevisionDojo

Question 1

A cost function, $C(x)$ , is defined by: $C(x) = \begin{cases} 100(0.5)^x & 0 \le x \le 4 \\ 6.25 & 4 < x \le 7 \\ 3x - 14.75 & x > 7 \end{cases}$

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

[5]

Question 2

Let

f(x)= \begin{cases} 2x+a, & x\le1, \\ x^2-3, & 1< x<4, \\ ax, & x\ge4. \end{cases}

Determine whether there exists a value of $a$ such that $f$ is continuous on $\mathbb{R}$ .

[5]

Question 3

Find the relationship between $a$ and $b$ such that the function $f(x)$ defined below is continuous at $x=1$ .

f(x) = \begin{cases} x^2 - ax + b, & x < 1 \\ ax - b, & x \ge 1 \end{cases}

[4]

Question 4

Find the relationship between the constants $c$ and $d$ such that

f(x) = \begin{cases} x^2 - 2x + c, & x \le 2 \\ 3x + d, & x > 2 \end{cases}

is continuous on $\mathbb{R}$ .

[5]

Question 5

Let $f(x)=\begin{cases}ax+1,&x<0,\\2x^2-3,&0\le x\le2,\\b x-4,&x>2.\end{cases}$

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

[5]

Question 6

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.

[3]

Question 7

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 $x > 7$ .

[6]

Question 8

A piecewise cost function $C(x)$ is defined for $x \ge 0$ based on the following properties:

For $0 \le x \le 4$ , the function decreases from $100$ to $6.25$ .
For $4 < x \le 7$ , the function remains constant at $6.25$ .
For $x > 7$ , the function increases linearly without bound.

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

[3]

Question 9

The cost model, $C(x)$ , is defined by the following piecewise function:

C(x) = \begin{cases} 100 \left( \frac{1}{2} \right)^x, & 0 \le x \le 4 \\ 6.25, & 4 < x \le 7 \\ 6.25 + 3(x - 7), & x > 7 \end{cases}

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

[4]

Question 10

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

[4]

Question 11

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

[4]

Question 12

Recall the piecewise cost function $C(x)$ defined as:

C(x) = \begin{cases} 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{cases}

Calculate $C(2)$ , $C(5)$ , and $C(10)$ .

[3]

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