Chapter 3

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=x^{3}\) of a cube when the edge lengths change from \(x_{0}\) to \(x_{0}+d x\)

Find the derivatives of the functions in Exercises \(23-50\). $$r=\sec \sqrt{\theta} \tan \left(\frac{1}{\theta}\right)$$

Find the first and second derivatives of the functions. $$s=\frac{t^{2}+5 t-1}{t^{2}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=t(t+1)(t+2)$$

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\sec ^{-1} 0\) b. \(\sin ^{-1} \sqrt{2}\)

Write a differential formula that estimates the given change in volume or surface area. The change in the surface area \(S=6 x^{2}\) of a cube when the edge lengths change from \(x_{0}\) to \(x_{0}+d x\)

Find the derivatives of the functions in Exercises \(23-50\). $$q=\sin \left(\frac{t}{\sqrt{t+1}}\right)$$

The line that is normal to the curve \(x^{2}+2 x y-3 y^{2}=0\) at (1,1) intersects the curve at what other point?

Find the limits. $$\lim _{x \rightarrow 2} \sin \left(\frac{1}{x}-\frac{1}{2}\right)$$

Find the first and second derivatives of the functions. $$r=\frac{(\theta-1)\left(\theta^{2}+\theta+1\right)}{\theta^{3}}$$

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

Which of the expressions are defined, and which are not? Give reasons for your answers. a. \(\cot ^{-1}(-1 / 2)\) b. \(\cos ^{-1}(-5)\)

Write a differential formula that estimates the given change in volume or surface area. The change in the lateral surface area \(S=\pi r \sqrt{r^{2}+h^{2}}\) of a right circular cone when the radius changes from \(r_{0}\) to \(r_{0}+d r\) and the height does not change

Find the derivatives of the functions in Exercises \(23-50\). $$q=\cot \left(\frac{\sin t}{t}\right)$$

Let \(p\) and \(q\) be integers with \(q>0 .\) If \(y=x^{p / q},\) differentiate the equivalent equation \(y^{4}=x^{p}\) implicitly and show that, for \(y \neq 0\) $$ \frac{d}{d x} x^{p / q}=\frac{p}{q} x^{(p / q)-1} $$

Find the limits. $$\lim _{x \rightarrow-\pi / 6} \sqrt{1+\cos (\pi \csc x)}$$

Find the first and second derivatives of the functions. $$u=\frac{\left(x^{2}+x\right)\left(x^{2}-x+1\right)}{x}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\frac{\theta+5}{\theta \cos \theta}$$

Use the identity $$\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u$$ to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 3.1 from the formula for the derivative of \(\sec ^{-1} u\)

Write a differential formula that estimates the given change in volume or surface area. The change in the volume \(V=\pi r^{2} h\) of a right circular cylinder when the radius changes from \(r_{0}\) to \(r_{0}+d r\) and the height does not change

Find the derivatives of the functions in Exercises \(23-50\). $$y=\cos \left(e^{-\theta^{2}}\right)$$

Show that if it is possible to draw three normals from the point \((a, 0)\) to the parabola \(x=y^{2}\) shown in the accompanying diagram, then \(a\) must be greater than \(1 / 2 .\) One of the normals is the \(x\) -axis. For what value of \(a\) are the other two normals perpendicular? (GRAPH CAN'T COPY)

Find the limits. $$\lim _{\theta \rightarrow \pi / 6} \frac{\sin \theta-\frac{1}{2}}{\theta-\frac{\pi}{6}}$$

Find the first and second derivatives of the functions. $$w=\left(\frac{1+3 z}{3 z}\right)(3-z)$$

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=-x^{2}$$

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

Use the identity $$\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u$$ to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 3.1 from the formula for the derivative of \(\sec ^{-1} u\)

Write a differential formula that estimates the given change in volume or surface area. The change in the lateral surface area \(S=2 \pi r h\) of a right circular cylinder when the height changes from \(h_{0}\) to \(h_{0}+d h\) and the radius does not change

Find the limits. $$\lim _{\theta \rightarrow \pi / 4} \frac{\tan \theta-1}{\theta-\frac{\pi}{4}}$$

Use a CAS to perform the following steps for the functions. a. Plot \(y=f(x)\) over the interval \(\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)\) b. Holding \(x_{0}\) fixed, the difference quotient $$q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$$ at \(x_{0}\) becomes a function of the step size \(h .\) Enter this function into your CAS workspace. c. Find the limit of \(q\) as \(h \rightarrow 0\) d. Define the secant lines \(y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)\) for \(h=3,2\) and \(1 .\) Graph them together with \(f\) and the tangent line over the interval in part (a). $$f(x)=x+\frac{5}{x}, \quad x_{0}=1$$

Find the first and second derivatives of the functions. $$p=\frac{q^{2}+3}{(q-1)^{3}+(q+1)^{3}}$$

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=-1 / x$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\frac{x \sqrt{x^{2}+1}}{(x+1)^{2 / 3}}$$

The radius of a circle is increased from 2.00 to \(2.02 \mathrm{m}\) a. Estimate the resulting change in area. b. Express the estimate as a percentage of the circle's original area.

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sin ^{2}(\pi t-2)$$

Verify that the following pairs of curves meet orthogonally. a. \(x^{2}+y^{2}=4, x^{2}=3 y^{2}\) b. \(x=1-y^{2}, \quad x=\frac{1}{3} y^{2}\)

Find the limits. $$\lim _{x \rightarrow 0} \sec \left[e^{x}+\pi \tan \left(\frac{\pi}{4 \sec x}\right)-1\right]$$

Use a CAS to perform the following steps for the functions. a. Plot \(y=f(x)\) over the interval \(\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)\) b. Holding \(x_{0}\) fixed, the difference quotient $$q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$$ at \(x_{0}\) becomes a function of the step size \(h .\) Enter this function into your CAS workspace. c. Find the limit of \(q\) as \(h \rightarrow 0\) d. Define the secant lines \(y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)\) for \(h=3,2\) and \(1 .\) Graph them together with \(f\) and the tangent line over the interval in part (a). $$f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2$$

Find the first and second derivatives of the functions. $$w=3 z^{2} e^{2 z}$$

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=x^{3} / 3$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt{\frac{(x+1)^{10}}{(2 x+1)^{5}}}$$

Use the identity $$\cot ^{-1} u=\frac{\pi}{2}-\tan ^{-1} u$$ to derive the formula for the derivative of \(\cot ^{-1} u\) in Table 3.1 from the formula for the derivative of \(\tan ^{-1} u\)

The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? The tree's cross-sectional area?

In Exercises \(51-70,\) find \(d y / d t\). $$y=\sec ^{2} \pi t$$

Find the limits. $$\lim _{x \rightarrow 0} \sin \left(\frac{\pi+\tan x}{\tan x-2 \sec x}\right)$$

Use a CAS to perform the following steps for the functions. a. Plot \(y=f(x)\) over the interval \(\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)\) b. Holding \(x_{0}\) fixed, the difference quotient $$q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}$$ at \(x_{0}\) becomes a function of the step size \(h .\) Enter this function into your CAS workspace. c. Find the limit of \(q\) as \(h \rightarrow 0\) d. Define the secant lines \(y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)\) for \(h=3,2\) and \(1 .\) Graph them together with \(f\) and the tangent line over the interval in part (a). $$f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi$$

Find the first and second derivatives of the functions. $$w=e^{-t}(z-1)\left(z^{2}+1\right)$$

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=x^{4} / 4$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sqrt[3]{\frac{x(x-2)}{x^{2}+1}}$$

What is special about the functions $$f(x)=\sin ^{-1} \frac{x-1}{x+1}, \quad x \geq 0, \quad \text { and } \quad g(x)=2 \tan ^{-1} \sqrt{x} ?$$ Explain.