91Ó°ÊÓ

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=x^{4}-8 x^{2}+2\)

Find the derivative of the function. $$ y=\sqrt{x}+\frac{1}{4} \sin (2 x)^{2} $$

Use a computer algebra system to differentiate the function. \(f(\theta)=\frac{\sin \theta}{1-\cos \theta}\)

Find the points at which the graph of the equation has a vertical or horizontal tangent line. $$ 4 x^{2}+y^{2}-8 x+4 y+4=0 $$

Find the derivative of the function. $$ y=\sin \sqrt[3]{x}+\sqrt[3]{\sin x} $$

Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function \(y=\frac{1+\csc x}{1-\csc x}\) Point \(\left(\frac{\pi}{6},-3\right)\)

Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] $$ \begin{aligned} &2 x^{2}+y^{2}=6 \\ &y^{2}=4 x \end{aligned} $$

Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. \(y=\frac{1}{x^{2}}\)

Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpendicular to each other.] $$ \begin{aligned} &y^{2}=x^{3} \\ &2 x^{2}+3 y^{2}=5 \end{aligned} $$

Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. Function \(f(x)=\tan x \cot x\) Point \((1,1)\)