Chapter 7

Evaluate the integrals. $$\int_{0}^{\pi / 12} 6 \tan 3 x d x$$

Evaluate the integrals. $$\int_{-\ln 4}^{-\ln 2} 2 e^{\theta} \cosh \theta d \theta$$

Find the limits $$\lim _{x \rightarrow \infty}(\ln x)^{1 / x}$$

Solve the initial value problems in Exercises \(51-54.\) $$\frac{d^{2} y}{d x^{2}}=2 e^{-x}, \quad y(0)=1 \quad \text { and } \quad y^{\prime}(0)=0$$

Evaluate the integrals. $$\int_{-1}^{-\sqrt{2} / 2} \frac{d y}{y \sqrt{4 y^{2}-1}}$$

Evaluate the integrals. $$\int \frac{d x}{2 \sqrt{x}+2 x}$$

Find the limits $$\lim _{x \rightarrow e^{+}}(\ln x)^{1 /(x-e)}$$

Evaluate the integrals. $$\int_{0}^{\ln 2} 4 e^{-\theta} \sinh \theta d \theta$$

Solve the initial value problems in Exercises \(51-54.\) $$\frac{d^{2} y}{d t^{2}}=1-e^{2 t}, \quad y(1)=-1 \quad \text { and } \quad y^{\prime}(1)=0$$

Evaluate the integrals. $$\int_{-2 / 3}^{-\sqrt{2} / 3} \frac{d y}{y \sqrt{9 y^{2}-1}}$$

Evaluate the integrals. $$\int \frac{\sec x d x}{\sqrt{\ln (\sec x+\tan x)}}$$

Find the limits $$\lim _{x \rightarrow 0^{+}} x^{-1 / \ln x}$$

Evaluate the integrals. $$\int_{\pi / 4}^{\pi / 4} \cosh (\tan \theta) \sec ^{2} \theta d \theta$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=2^{x}$$

If \(f(x)\) is one-to-one, can anything be said about \(g(x)=-f(x) ?\) Is it also one-to-one? Give reasons for your answer.

Evaluate the integrals. $$\int \frac{3 d r}{\sqrt{1-4(r-1)^{2}}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{x(x+1)}$$

Find the limits $$\lim _{x \rightarrow \infty} x^{1 / \ln x}$$

Evaluate the integrals. $$\int_{0}^{\pi / 2} 2 \sinh (\sin \theta) \cos \theta d \theta$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=3^{-x}$$

If \(f(x)\) is one-to-one and \(f(x)\) is never zero, can anything be said about \(h(x)=1 / f(x) ?\) Is it also one-to-one? Give reasons for your answer.

Evaluate the integrals. $$\int \frac{6 d r}{\sqrt{4-(r+1)^{2}}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\left(x^{2}+1\right)(x-1)^{2}}$$

Find the limits $$\lim _{x \rightarrow \infty}(1+2 x)^{1 /(2 \ln x)}$$

Evaluate the integrals. $$\int_{1}^{2} \frac{\cosh (\ln t)}{t} d t$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=5^{\sqrt{s}}$$

Suppose that the range of \(g\) lies in the domain of \(f\) so that the composite \(f \circ g\) is defined. If \(f\) and \(g\) are one-to-one, can anything be said about \(f \circ g ?\) Give reasons for your answer.

Evaluate the integrals. $$\int \frac{d x}{2+(x-1)^{2}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\frac{t}{t+1}}$$

Find the limits $$\lim _{x \rightarrow 0}\left(e^{x}+x\right)^{1 / x}$$

Evaluate the integrals. $$\int_{1}^{4} \frac{8 \cosh \sqrt{x}}{\sqrt{x}} d x$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=2^{\left(s^{2}\right)}$$

If a composite \(f \circ g\) is one-to-one, must \(g\) be one-to-one? Give reasons for your answer.

Evaluate the integrals. $$\int \frac{d x}{1+(3 x+1)^{2}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\frac{1}{t(t+1)}}$$

Find the limits $$\lim _{x \rightarrow 0^{+}} x$$

Evaluate the integrals. $$\int_{-\ln 2}^{0} \cosh ^{2}\left(\frac{x}{2}\right) d x$$

Assume that \(f\) and \(g\) are differentiable functions that are inverses of one another so that \((g \circ f)(x)=x .\) Differentiate both sides of this equation with respect to \(x\) using the Chain Rule to express \((g \circ f)^{\prime}(x)\) as a product of derivatives of \(g\) and \(f .\) What do you find? (This is not a proof of Theorem 1 because we assume here the theorem's conclusion that \(g=f^{-1}\) is differentiable.)

Evaluate the integrals. $$\int \frac{d x}{(2 x-1) \sqrt{(2 x-1)^{2}-4}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\theta+3} \sin \theta$$

Find the limits $$\lim _{x \rightarrow 0^{+}}\left(1+\frac{1}{x}\right)^{x}$$

Evaluate the integrals. $$\int_{0}^{\ln 10} 4 \sinh ^{2}\left(\frac{x}{2}\right) d x$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=t^{1-e}$$

Evaluate the integrals. $$\int \frac{d x}{(x+3) \sqrt{(x+3)^{2}-25}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\tan \theta) \sqrt{2 \theta+1}$$

When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown here. (We give one derivation in Section 8.4.) $$\begin{aligned}&\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1}), \quad-\infty < x<\infty\\\&\cosh ^{-1} x=\ln (x+\sqrt{x^{2}-1}), \quad x \geq 1\\\ &\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}, \quad|x| < 1\\\&\operatorname{sech}^{-1} x=\ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right), \quad0 < x \leq 1\\\&\operatorname{csch}^{-1} x=\ln \left(\frac{1}{x}+\frac{\sqrt{1+x^{2}}}{|x|}\right), \quad x \neq 0\\\&\operatorname{coth}^{-1} x=\frac{1}{2} \ln \frac{x+1}{x-1}, \quad |x|>1\end{aligned}$$ Use the formulas in the box here to express the numbers in terms of natural logarithms. $$\sinh ^{-1}(-5 / 12)$$

Find the limits $$\lim _{x \rightarrow \infty}\left(\frac{x+2}{x-1}\right)^{x}$$

Find the derivative of \(y\) with respect to the given independent variable. $$y=(\cos \theta)^{\sqrt{2}}$$

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\) c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g,\) the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right) .\) Discuss the symmetries you see across the main diagonal. $$y=\sqrt{3 x-2}, \quad \frac{2}{3} \leq x \leq 4, \quad x_{0}=3$$

Evaluate the integrals. $$\int_{-\pi / 2}^{\pi / 2} \frac{2 \cos \theta d \theta}{1+(\sin \theta)^{2}}$$