91Ó°ÊÓ

Find \(y^{\prime \prime \prime}\) for each function. $$y=\frac{1}{1-x}$$

Find \(y^{\prime \prime \prime}\) for each function. $$y=x \sqrt{1+x^{2}}$$

Find \(\frac{d y}{d x}\) for each pair of functions.\( Find an equation for the tangent line to the graph of \)y=\left(\frac{2 x+3}{x-1}\right)^{3}$ at the point (2,343).

Is the function given by \(f(x)=\left\\{\begin{array}{ll}\frac{x^{2}-4 x-5}{x-5} & \text { for } x<5, \\ x+1 & \text { for } x \geq 5.\end{array}\right.\) continuous at \(x=5 ?\) Why or why not?

Find \(y^{\prime \prime \prime}\) for each function. $$y=\frac{1}{\sqrt{2 x+1}}$$

Consider $$g(x)=\left(\frac{6 x+1}{2 x-5}\right)^{2}$$ a) Find \(g^{\prime}(x)\) using the Extended Power Rule. b) Note that \(g(x)=\frac{36 x^{2}+12 x+1}{4 x^{2}-20 x+25}\) Find \(g^{\prime}(x)\) using the Quotient Rule. c) Compare your answers to parts (a) and (b). Which approach was easier, and why?

Find \(y^{\prime \prime \prime}\) for each function. $$y=\frac{3 x-1}{2 x+3}$$

Is the function given by \(G(x)=\left\\{\begin{array}{ll}\frac{x^{2}-3 x-4}{x-4} & \text { for } x<4, \\ 2 x-3 & \text { for } x \geq 4.\end{array}\right.\) continuous at \(x=4 ?\) Why or why not?

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$y=x^{2}-3$$

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$h(x)=\left(3 x^{2}-7\right)^{5}$$