Calculus › Continuity

Continuity of piecewise functions

Solve the first problem 11 problems

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1. Continuity of a piecewise linear function ●○○○○ 10 ptsLet $f(n) = 4n+3$ if $n<a$, and $f(n) = 7n-12$ if $n\ge a$. Find $a$ so that the graph of $y=f(n)$ is continuous.
2. Piecewise function coefficients sum ●○○○○ 10 ptsLet $f(x) = \left\{ \begin{array}{cl} ax+3 & \text{ if }x>0, \\ ab & \text{ if }x=0, \\ bx+c & \text{ if }x<0. \end{array} \right.$ If $f(2)=5$, $f(0)=5$, and $f(-2)=-10$, and $a$, $b$, and $c$ are nonnegative integers, then find $a+b+c$.
3. Continuity constant for a piecewise function ●○○○○ 10 ptsLet $f(x)=\begin{cases} kx^{2} & \text{if } x\le 2 \\ x+k & \text{if } x>2\end{cases}$. Find the value of the constant $k$ that makes $f$ continuous. Round to two decimal places.
4. Continuity of a piecewise function ●●○○○ 20 ptsLet $f(x) = \begin{cases} 5x^2+2 & \text{if } x\le a, \\ 11x & \text{if } x>a. \end{cases}$ Find the smallest possible value of $a$ for which the graph of $y=f(x)$ is continuous (the graph can be drawn without lifting your pencil from the paper). Give your answer as a decimal.
5. Making a piecewise function differentiable at a join ●●○○○ 20 ptsLet $f(x)=ax+b$ for $x\le 2$ and $f(x)=2-x^2$ for $x>2$. Find $a$ and $b$ so that $f$ is differentiable at $x=2$.
6. Continuity with parameters a and b ●●○○○ 20 ptsLet $f,g:\mathbb{R}\to\mathbb{R}$ be the functions $$f(x)=\begin{cases}-x+3,&x\leq 1,\\-ax^2,&\text{otherwise},\end{cases}\qquad g(x)=\begin{cases}\dfrac{x^3+x^2-x-1}{x-1},&x\neq 1,\\-b,&\text{otherwise}.\end{cases}$$ Determine the parameters $a,b\in\mathbb{R}$ such that $f$ and $g$ are both continuous. Find $a$ and $b$.
7. Piecewise function with equal values ●●●○○ 50 ptsLet $a \neq 0$ be a real number, and define the function $f(x)$ by $$f(x)=\begin{cases}2x+a, & x<1,\\ -x-2a, & x\geq 1.\end{cases}$$ If $f(1-a)=f(1+a)$, find the value of $a$. Give your answer as a decimal.

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