Chapter 29

Logarithmic Functions $$\int_{2}^{4} x \log \left(x^{2}+1\right) d x$$

Find \(d y / d x\) for each implicit function. $$y \tan x=2$$

Implicit Relations Find \(d y / d x\) $$x y=a^{2} \ln \frac{x}{a}$$

Implicit Functions Find \(d y / d x\) for each implicit function. $$y \sin x=1$$

Find the area under each curve.$$y=2 \cos x \quad \text { from } x=-\pi / 2 \text { to } \pi / 2$$

With Trigonometric Functions. Differentiate. $$y=e^{x} \sin x$$

Logarithmic Functions $$\int_{1}^{3} x^{2} \log \left(2+3 x^{3}\right) d x$$

Find \(d y / d x\) for each implicit function. $$x y+y \cot x=0$$

Implicit Relations Find \(d y / d x\) $$\ln y+x=10$$

Implicit Functions Find \(d y / d x\) for each implicit function. $$x y-y \sin x-x \cos y=0$$

Find the area under the curve \(y=e^{2 x}\) from \(x=1\) to 3

With Trigonometric Functions. Differentiate. $$y=e^{\theta} \cos 2 \theta$$

Find \(d y / d x\) for each implicit function. $$\sec (x+y)=7$$

Logarithmic Differentiation Differentiate. Remember to start these by taking the logarithm of both sides. $$y=\frac{\sqrt{x+2}}{\sqrt[3]{2-x}}$$

Implicit Functions Find \(d y / d x\) for each implicit function. $$y=\cos (x-y)$$

Find the area under each curve.Find the area between the curve \(y=\sin x\) and the \(x\) axis from \(x=1\) rad to 3 rad.

With Trigonometric Functions. Differentiate. $$y=e^{x}(\cos b x+\sin b x)$$

Find \(d y / d x\) for each implicit function. $$x \cot y=y \sec x$$

Logarithmic Differentiation Differentiate. Remember to start these by taking the logarithm of both sides. $$y=\frac{\sqrt{a^{2}-x^{2}}}{x}$$

Find the area under each curve.Find the area between the curve \(y=\cos x\) and the \(x\) axis from \(x=0\) to \(\frac{3}{2} \pi\).

The first-quadrant area bounded by \(y=e^{x}\) and \(x=1\) is rotated about the line \(x=1 .\) Find the volume generated.

Implicit Relations. Find \(d y / d x\). $$e^{x}+e^{y}=1$$

Find the tangent to the curve \(y=\tan x\) at \(x=1\) rad.

Logarithmic Differentiation Differentiate. Remember to start these by taking the logarithm of both sides. $$y=x^{x}$$

Implicit Functions Find \(d y / d x\) for each implicit function. $$x \sin y-y \sin x=0$$

Find the length of the catenary \(y=(a / 2)\left(e^{x / a}+e^{-x / a}\right)\) from \(x=0\) to \(6 .\) Use \(a=3\)

Find the tangent to the curve \(y=\sec 2 x\) at \(x=2\) rad.

Logarithmic Differentiation Differentiate. Remember to start these by taking the logarithm of both sides. $$y=x^{\sin x}$$

Tangents Find the slope of the tangent to four significant digits at the given value of \(x .\) $$y=\sin x \quad \text { at } x=2 \text { rad }$$

The curve \(y=e^{-x}\) is rotated about the \(x\) axis. Find the area of the surface generated, from \(x=0\) to 100

Implicit Relations. Find \(d y / d x\). $$e^{y}=\sin (x+y)$$

Logarithmic Differentiation Differentiate. Remember to start these by taking the logarithm of both sides. $$y=(\cot x)^{\sin x}$$

Evaluate each expression. $$f^{\prime}(2) \text { where } f(x)=e^{\sin (\pi x / 2)}$$

Find the horizontal distance \(\bar{x}\) to the centroid of the area formed by the curve \(y=\frac{1}{2}\left(e^{x}+e^{-x}\right),\) the coordinate axes, and the line \(x=1\)

For each function, find any maximum, minimum, or inflection points between 0 and \(\pi.\) $$y=\tan x-4 x$$

Tangents Find the slope of the tangent to four significant digits at the given value of \(x .\) $$y=x \sin \frac{x}{2} \quad \text { at } x=2 \text { rad }$$

Find the area under each curve.Find the coordinates of the centroid of the area bounded by the \(x\) axis and a half-cycle of the sine curve \(y=\sin x\).

Evaluate each expression. $$f^{\prime \prime}(0) \text { where } f(t)=e^{\sin t} \cos t$$

Find the vertical distance \(\bar{y}\) to the centroid of the area formed by the curve \(y=e^{x}\) between \(x=0\) and 1

An object moves with simple harmonic motion so that its displacement \(y\) at time \(t\) is \(y=6 \sin 4 t \mathrm{cm} .\) Find the velocity and acceleration of the object when \(t=0.0500 \mathrm{s}.\)

Tangent to a Curve Find the slope of the tangent at the point indicated. $$y=\log x \quad \text { at } x=1$$

Tangents Find the slope of the tangent to four significant digits at the given value of \(x .\) $$y=\sin x \cos 2 x \quad \text { at } x=1 \text { rad }$$

Evaluate each expression. $$f^{\prime}(1) \text { and } f^{\prime \prime}(1) \text { where } f(x)=e^{x-1} \sin \pi x$$

Extreme Values and Inflection Points For each curve, find the maximum, minimum, and inflection points between \(x=0\) and \(2 \pi\). $$y=\sin x$$

The current at a point in a certain circuit is given by $$i=84.3 \sin (11.5 t+5.48) \mathrm{A}$$ (a) Write an expression for the charge at that point, assuming an initial charge of \(0,\) and (b) evaluate it at \(t=2.00 \mathrm{s}\).

With Logarithmic Functions. Differentiate. $$y=e^{x} \ln x$$

Find the moment of inertia of the area bounded by the curve \(y=e^{x},\) the line \(x=1,\) and the coordinate axes, with respect to the \(x\) axis.

Tangent to a Curve Find the slope of the tangent at the point indicated. $$y=\ln \left(x^{2}+2\right) \text { at } x=4$$

Extreme Values and Inflection Points For each curve, find the maximum, minimum, and inflection points between \(x=0\) and \(2 \pi\). $$y=\frac{x}{2}-\sin x$$

An expression for the current at a point in a certain circuit is $$i=273 \sin (382 t+0.573) \mathrm{A}$$. (a) Assuming an initial charge of \(0,\) write an expression for the charge at that point and (b) evaluate it at \(t=3.50 \mathrm{s}\).