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Exercises for Question Type 8: Determining continuity of piecewise models - IB | RevisionDojo

Question 1

Express $b$ in terms of $a$ so that the function

f(x)=\begin{cases} x^2, & x \le 2 \\ ax+b, & x > 2 \end{cases}

is continuous at $x=2$ .

[5]

Question 2

Determine $a$ so that $f(x)=\begin{cases} \sqrt{x+3},&-3\le x<1,\\ ax^2+1,&x\ge1 \end{cases}$ is continuous at $x=1$ .

[4]

Question 3

Determine $b$ so that

f(x)=\begin{cases} 3x+b,&x<2,\\ 5x+1,&x\ge2 \end{cases}

is continuous at $x=2$ .

[4]

Question 4

Find $k$ so that

f(x)=\begin{cases} \dfrac{\sin x}{x},&x\neq0,\\ k,&x=0 \end{cases}

is continuous at $x=0$ .

[3]

Question 5

Find the value of $a$ such that the function

f(x) = \begin{cases} x^2 + ax + 1, & x \le 1 \\ 2ax + 1, & x > 1 \end{cases}

is continuous at $x = 1$ .

[3]

Question 6

Find $a$ so that

f(x) = \begin{cases} x^3 - 8, & x < 2, \\ a(x - 2), & x \ge 2 \end{cases}

is continuous at $x = 2$ .

[3]

Question 7

Find $a$ so that the function

f(x) = \begin{cases} \dfrac{x^2 - 1}{x - 1}, & x \neq 1 \\ a, & x = 1 \end{cases}

is continuous at $x = 1$ .

[3]

Question 8

Calculus: Continuity of piecewise functions

Find the value of $a$ such that the function

f(x)=\begin{cases} x+2, & x < 1 \\ a x-3, & x \ge 1 \end{cases}

is continuous at $x=1$ .

[3]

Question 9

Given the piecewise function

f(x)=\begin{cases} ax+b, & x<1, \\ 2x+1, & 1\le x<3, \\ cx-1, & x\ge3, \end{cases}

find $b$ and $c$ in terms of $a$ so that $f$ is continuous at both $x=1$ and $x=3$ .

[6]

Question 10

Determine $m$ so that

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

is continuous at $x=1$ .

[4]

Question 11

Limits and continuity of piecewise functions.

Determine the value of $a$ so that

f(x) = \begin{cases} x+2, & x < 0 \\ a x^2+2, & 0 \le x < 1 \\ 2x+1, & x \ge 1 \end{cases}

is continuous on its domain.

[3]

Question 12

Determine $k$ such that

f(x)=\begin{cases} kx+3, & x<3 \\ 2x-1, & x\ge 3 \end{cases}

is continuous at $x=3$ .

[3]

Question Type 8: Determining continuity of piecewise models Exercises