91Ó°ÊÓ

In Exercises 39-42, find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=\frac{1+2 x^{1 / 2}}{1+x^{3 / 2}} ;\left(4, \frac{5}{9}\right)\)

Find the derivative of the function. \(g(t)=\frac{(2 t-1)^{2}}{(3 t+2)^{4}}\)

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=|x-1| $$

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes time and \(f(x)\) denotes the prime interest rate at time \(x\).

Find the indicated limit given that \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=4\) \(\lim _{x \rightarrow a} 2 f(x)\)

Find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=-\frac{5}{3} x^{2}+2 x+2 ;\left(-1,-\frac{5}{3}\right)\)

Find the slope and an equation of the tangent line to the graph of the function \(f\) at the specified point. \(f(x)=x^{4}-3 x^{3}+2 x^{2}-x+1 ;(1,0)\)

In Exercises 43-48, find the first and second derivatives of the given function. \(f(x)=4 x^{2}-2 x+1\)

Find the derivative of the function. \(f(x)=\frac{\sqrt{2 x+1}}{x^{2}-1}\)

Determine the values of \(x\), if any, at which each function is discontinuous. At each number where \(f\) is discontinuous, state the condition(s) for continuity that are violated. $$ f(x)=\left\\{\begin{array}{ll} x+5 & \text { if } x<0 \\ 2 & \text { if } x=0 \\ -x^{2}+5 & \text { if } x>0 \end{array}\right. $$