Chapter 4

Differentiate the functions with respect to the independent variable. \(f(x)=x^{2} e^{-x}\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=x^{2} \sin \frac{\pi}{3}+\tan \frac{\pi}{4} $$

In the following examples quantities \(x\) and \(y\) are given. Interpret the role of change dy/dx in words.11. A car moves along a straight road. Its location at time \(t\) is given by $$ s(t)=10 t^{2}, 0 \leq t \leq 2 $$ where \(t\) is measured in hours and \(s(t)\) is measured in kilometers.

Find the first 10 derivatives of \(y=x^{6}\).

In Problems 9-11, find the lines that are (a) tangential and (b) normal to each curve at the given point. $$ \frac{x^{2}}{25}-\frac{y^{2}}{9}=1,\left(\frac{25}{3}, 4\right) \text { (hyperbola) } $$

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{3 x+1}{\sqrt{2 x^{2}-1}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{1-x}\) at \(a=0\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=(x+3)^{2}\)

Find the derivative with respect to the independent variable. $$ f(x)=\cot (2-3 x) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\left(3 x^{2}-1\right) e^{1-x^{2}}\)

Use the product rule to find the derivative with respect to the independent variable. \(f(x)=\frac{\left(2 x^{2}-3 x+1\right)^{2}}{4}+2\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=2 x^{3} \cos \frac{\pi}{3}+\cos \frac{\pi}{6} $$

A train moves along a straight line. Its location at time \(t\) is given by $$ s(t)=\frac{100}{t}, \quad 1 \leq t \leq 5 $$ where \(t\) is measured in hours and \(s(t)\) is measured in kilometers. (a) Graph \(s(t)\) for \(1 \leq t \leq 5\). (b) Find the average velocity of the train between \(t=1\) and \(t=5\). Where on the graph of \(s(t)\) can you find the average velocity? (c) Use calculus to find the instantaneous velocity of the train at \(t=2 .\) Where on the graph of \(s(t)\) can you find the instantaneous velocity? What is the speed of the train at \(t=2 ?\)

Find \(f^{(n)}(x)\) and \(f^{(n+1)}(x)\) if \(f(x)=x^{n}\).

(a) The curve with equation \(y^{2}=x^{2}-x^{4}\) is shaped like the numeral eight. Find \(\frac{d y}{d x}\) at \(\left(\frac{1}{2}, \frac{1}{4} \sqrt{3}\right)\). (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately; that is, graph $$ \begin{array}{l} y_{1}=\sqrt{x^{2}-x^{4}} \\ y_{2}=-\sqrt{x^{2}-x^{4}} \end{array} $$ Choose the viewing rectangle \(-2 \leq x \leq 2,-1 \leq y \leq 1\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\left(1-3 x^{2}\right)^{3}}{\left(3-x^{2}\right)^{2}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{2}{1+x}\) at \(a=1\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=x^{2}+6 x+9\)

Find the derivative with respect to the independent variable. $$ f(x)=2 \sec (1+2 x) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{1+e^{x}}{1+x^{2}}\)

Use the product rule to find the derivative with respect to the independent variable. \(g(s)=\left(2 s^{2}-5 s\right)^{2}\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=-3 x^{4} \tan \frac{\pi}{6}-\cot \frac{\pi}{6} $$

If \(s(t)\) denotes the position of an object that moves along a straight line, then \(\Delta s / \Delta t\), called the average velocity, is the average rate of change of \(s(t)\), and \(v(t)=d s / d t\), called the (instantaneous) velocity, is the instantaneous rate of change of \(s(t) .\) The speed of the object is the absolute value of the velocity, \(|v(t)|\). Suppose now that a car moves along a straight road. The location at time \(t\) is given by $$ s(t)=\frac{160}{3} t^{2}, \quad 0 \leq t \leq 1 $$ where \(t\) is measured in hours and \(s(t)\) is measured in kilometers. (a) Where is the car at \(t=3 / 4\), and where is it at \(t=1 ?\) (b) Find the average velocity of the car between \(t=3 / 4\) and \(t=1\) (c) Find the velocity and the speed of the car at \(t=3 / 4\).

Find a second-degree polynomial \(p(x)=a x^{2}+b x+c\) with \(p(0)=3, p^{\prime}(0)=2\), and \(p^{\prime \prime}(0)=6 .\)

(a) Consider the curve with equation \(x^{2 / 3}+y^{2 / 3}=4\). Find \(\frac{d y}{d x}\) at \((-1,3 \sqrt{3})\) (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately. To get the left half of the graph, make sure that your calculator evaluates \(x^{2 / 3}\) in the order \(\left(x^{2}\right)^{1 / 3}\). Choose the viewing rectangle \(-10 \leq x \leq 10\), \(-10 \leq y \leq 10 .\)

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\sqrt{2 x-1}}{(x-1)^{2}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{3-2 x}\) at \(a=2\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=x^{2}+4 x+4\)

Find the derivative with respect to the independent variable. $$ f(x)=-3 \csc (3-5 x) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{x-e^{-x}}{1+x e^{-x}}\)

Use the product rule to find the derivative with respect to the independent variable. \(h(t)=4\left(3 t^{2}-1\right)(2 t+1)\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=x^{2} \sec \frac{\pi}{6}+3 x \sec \frac{\pi}{4} $$

Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds and \(s(t)\) is measured in meters. (a) Graph \(s(t)\) for \(t \geq 0\). (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time \(0 ?\) (ii) Is there another time at which the particle visits the location where it was at time 0 ? (iii) How far to the right on the straight line does the particle travel? (iv) How far to the left on the straight line does the particle travel? (v) Where is the velocity positive? negative? equal to 0 ? (c) Find the velocity of the particle. (d) When is the velocity of the particle equal to \(1 \mathrm{~m} / \mathrm{s}\) ?

14\. Velocity Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds and \(s(t)\) is measured in meters. (a) Graph \(s(t)\) for \(t \geq 0\). (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time \(0 ?\) (ii) Is there another time at which the particle visits the location where it was at time \(0 ?\) (iii) How far to the right on the straight line does the particle travel? (iv) How far to the left on the straight line does the particle travel? (v) Where is the velocity positive? negative? equal to \(0 ?\) (c) Find the velocity of the particle. (d) When is the velocity of the particle equal to \(1 \mathrm{~m} / \mathrm{s}\) ?

The position at time \(t\) of a particle that moves along a straight line is given by the function \(s(t) .\) The first derivative of \(s(t)\) is called the velocity, denoted by \(v(t) ;\) that is, the velocity is the rate of change of the position. The rate of change of the velocity is called acceleration, denoted by \(a(t) ;\) that is, $$\frac{d}{d t} v(t)=a(t) $$ Given that \(v(t)=s^{\prime}(t)\), it follows that $$\frac{d^{2}}{d t^{2}} s(t)=a(t) $$ Find the velocity and the acceleration at time \(t=1\) for the following position functions: (a) \(s(t)=t^{2}-3 t\) (b) \(s(t)=\sqrt{t^{2}+1}\) (c) \(s(t)=t^{4}-2 t\).

(a) Consider the curve with equation \(y^{2}=10 x^{4}-x^{2}\). Find \(\frac{d y}{d x}\) at \((1,3)\) (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately. Choose the viewing rectangle \(-3 \leq x \leq 3,-10 \leq y \leq 10\)

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\sqrt{x^{2}+1}}{2+\sqrt{x^{2}+1}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{(1+x)^{2}}\) at \(a=0\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(f(x)=\sin \left(\frac{\pi}{2} x\right)\)

Find the derivative with respect to the independent variable. $$ f(x)=3 \sin \left(x^{2}\right) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{e^{x}+e^{-x}}{2+e^{x}}\)

Use the product rule to find the derivative with respect to the independent variable. \(g(t)=3\left(2 t^{2}-5 t^{4}\right)^{2}\)

Differentiate the functions given with respect to the independent variable. $$ f(t)=t^{3} e^{-2}+t+e^{-1} $$

Neglecting air resistance, the height \(h\) (in meters) of an object thrown vertically from the ground with initial velocity \(v_{0}\) is given by $$h(t)=v_{0} t-\frac{1}{2} g t^{2} $$ where \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\) is the earth's gravitational constant and \(t\) is the time (in seconds) elapsed since the object was released. (a) Find the velocity and the acceleration of the object. (b) Find the time when the velocity is equal to 0 . In which direction is the object traveling right before this time? in which direction right after this time?

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(s)=\sqrt[n]{s+\sqrt[n]{s}} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\frac{1}{(1-x)^{2}}\) at \(a=0\)

Find \(c\) so that \(f^{\prime}(c)=0 .\) \(\cos (\pi-x)\)

Find the derivative with respect to the independent variable. $$ f(x)=2 \cos \left(x^{3}-3 x\right) $$

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{x}{e^{x}+e^{-x}}\)

Use the product rule to find the derivative with respect to the independent variable. \(h(s)=\left(4-3 s^{2}+4 s^{3}\right)^{2}\)