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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. 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.
3. 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.
4. 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 $b$.
5. Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$. Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$. Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$.
6. Proportion exceeding a rounded threshold ●●●○○ 50 ptsThe tensile strength of a metal component is normally distributed with mean $10000$ kg/cm² and standard deviation $100$ kg/cm². Each measurement is recorded to the nearest $50$ kg/cm². A measurement counts as exceeding $10150$ when its recorded value is greater than $10150$. What proportion of the measurements exceed $10150$ kg/cm²? Round to four decimal places.