91Ó°ÊÓ

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. \(y=\sinh ^{-1}(\tan x)\)

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{y \rightarrow 0} \frac{\sqrt{5 y+25}-5}{y}$$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\int_{x^{2} / 2}^{x^{2}} \ln \sqrt{t} d t$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\csc ^{-1}\left(e^{t}\right) $$

Evaluate the integrals. $$\int_{0}^{\ln 16} e^{x / 4} d x$$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\int_{\sqrt{x}}^{\sqrt[3]{x}} \ln t d t$$

\(\begin{array}{l}{\text { a. Find } f^{-1}(x)} \\ {\text { b. Graph } f \text { and } f^{-1} \text { together. }} \\ {\text { c. Evaluate } d f / d x \text { at } x=a \text { and } d f^{-1} / d x \text { at } x=f(a) \text { to show that }} \\ \quad {\text { at these points } d f^{-1} / d x=1 /(d f / d x) .}\end{array}\) \(f(x)=(1 / 5) x+7, \quad a=-1\)

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. \(y=\cosh ^{-1}(\sec x), \quad 0

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\cos ^{-1}\left(e^{-t}\right) $$

Continuous price discounting To encourage buyers to place 100 -unit orders, your firm's sales department applies a continuous discount that makes the unit price a function \(p(x)\) of the number of units \(x\) ordered. The discount decreases the price at the rate of \(\$ 0.01\) per unit ordered. The price per unit for a 100 -unit order is \(p(100)=\$ 20.09 .\) a. Find \(p(x)\) by solving the following initial value problem: $$ \begin{aligned} \text { Differential equation: } & \frac{d p}{d x}=-\frac{1}{100} p \\ \text { Initial condition: } & p(100)=20.09 \end{aligned} $$ b. Find the unit price \(p(10)\) for a 10 -unit order and the unit price \(p(90)\) for a 90 -unit order. c. The sales department has asked you to find out if it is discounting so much that the firm's revenue, \(r(x)=x \cdot p(x),\) will actually be less for a 100 -unit order than, say, for a 90-unit order. Reassure them by showing that \(r\) has its maximum value at \(x=100 .\) d. Graph the revenue function \(r(x)=x p(x)\) for \(0 \leq x \leq 200\)