Find the value of the constant $b$ such that the function $f(x)=\sqrt{x+1}+\frac{b}{x}$ has a point of inflection at $x=3$.

Use the algorithm for curve sketching to sketch the graph of each function.

(a). $f(x)=\frac{3-x}{2 x+5}$

(b). $h(t)=2 t^3-15 t^2+36 t-10$

($c$). $y=\frac{20}{x^2+4}$

(d). $s(t)=t+\frac{1}{t}$

(e). $g(x)=\frac{2 x^2+5 x+2}{x+3}$

(f). $s(t)=\frac{t^2+4 t-21}{t-3}, t \geq-7$

Find constants a, b, and c such that the function $f(x)=a x^3+b x^2+c$ will have a local extremum at $(2,11)$ and a point of inflection at $(1,5)$. Sketch the graph of $y=f(x)$.

Determine values of a, b, and c such that the graph of $y=a x^2+b x+c$ has a relative maximum at $(3,12)$ and crosses the $y$-axis at $(0,1)$.

Sketch each function using suitable techniques.

(a). $y=x^4-8 x^2+7$

(b). $f(x)=\frac{3 x-1}{x+1}$

($c$). $g(x)=\frac{x^2+1}{4 x^2-9}$

(d). $y=x(x-4)^3$

(e). $h(x)=\frac{x}{x^2-4 x+4}$

(f). $f(t)=\frac{t^2-3 t+2}{t-3}$

For each of the following, check for discontinuities and state the equation of any vertical asymptotes. Conduct a limit test to determine the behaviour of the curve on either side of the asymptote.

(a). $f(x)=\frac{x}{(x-5)^2}$

(b). $f(x)=\frac{5}{x^2+9}$

($c$). $f(x)=\frac{x-2}{x^2-12 x+12}$

Use a graphing calculator or graphing software to graph each of the following functions. Inspect the graph to determine where the function is increasing and where it is decreasing.

(a). $y=3 x^2-12 x+7$

(b). $y=4 x^3-12 x^2+8$

($c$). $f(x)=\frac{x+2}{x+3}$

(d). $f(x)=\frac{x^2-1}{x^2+3}$

Solve each equation.

(a). $2 y^2+y-3=0$

(b). $x^2-5 x+3=17$

($c$). $4 x^2+20 x+25=0$

(d). $y^3+4 y^2+y-6=0$

If a polynomial function of degree three has a local minimum, explain how the function's values behave as $x \rightarrow+\infty$ and as $x \rightarrow-\infty$. Consider all cases.

Determine the points at which $f^{\prime}(x)=0$ for each of the following functions:

(a). $f(x)=x^3+6 x^2+1$

(b). $f(x)=\sqrt{x^2+4}$

($c$). $f(x)=(2 x-1)^2\left(x^2-9\right)$

(d). $f(x)=\frac{5 x}{x^2+1}$

Make a rough sketch of the graph, and answer the questions:

When would you recommend that the owner sell her shares? Label this point $S$ on your graph. What do you notice about the slope, or instantaneous rate of change, of the graph at this point?

Explain what it means to determine the critical points of the graph of a given function.

For an object moving with constant negative acceleration, draw the following: a graph of position vs. time

Use the precise definition of a limit to prove that $\lim_{(x,y)\to (a,b)}c=c$ where $c$ is a constant.