Chapter 4

In Problems , find the normal line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=\sqrt{3} x^{4}-2 \sqrt{3} x^{2} \text { , at } x=-\sqrt{3} $$

Explain the relationship between continuity and differentiability.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\left(2 x^{3}-x\right) \cos \left(1-x^{2}\right) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ g(r)=4^{r^{1 / 4}} $$

Differentiate with respect to the independent variable. $$ f(x)=\frac{x^{4}+2 x-1}{5 x^{2}-2 x+1} $$

Differentiate the functions with respect to the independent variable. $$ h(s)=\ln (\ln s) $$

In Problems , find the normal line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=-2 x^{2}-x, \text { at } x=0 $$

Sketch the graph of a function that is continuous at all points in its domain and differentiable in the domain except at one point.

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\sin (2 x)}{1+x^{2}} $$

Compute the limits in Problems \(53-56 .\) $$ \lim _{h \rightarrow 0} \frac{e^{2 h}-1}{h} $$

Differentiate with respect to the independent variable. $$ f(x)=\frac{3-x^{3}}{1-x} $$

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

Sketch the graph of a periodic function defined on \(\mathbf{R}\) that is continuous at all points in its domain and differentiable in the domain except at \(c=k, k \in \mathbf{Z}\).

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{1+\cos (3 x)}{2 x^{3}-x} $$

Compute the limits in Problems \(53-56 .\) $$ \lim _{h \rightarrow 0} \frac{e^{5 h}-1}{3 h} $$

Differentiate with respect to the independent variable. $$ f(x)=\frac{1+2 x^{2}-4 x^{4}}{3 x^{3}-5 x^{5}} $$

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

In Problems , find the normal line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=1-\pi x^{2}, \text { at } x=-1 $$

If \(f(x)\) is differentiable for all \(x \in \mathbf{R}\) except at \(x=c\), is it true that \(f(x)\) must be continuous at \(x=c\) ? Justify your answer.

Find the lines that are \((a)\) tangential and \((b)\) normal to each curve at the given point. \(x^{2}+y^{2}=25,(4,-3)\) (circle)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\tan \frac{1}{x} $$

Compute the limits in Problems \(53-56 .\) $$ \lim _{h \rightarrow 0} \frac{e^{h}-1}{\sqrt{h}} $$

Differentiate with respect to the independent variable. $$ h(t)=\frac{t^{2}-3 t+1}{t+1} $$

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

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=|x-2| $$

Find the lines that are \((a)\) tangential and \((b)\) normal to each curve at the given point. \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1,\left(1, \frac{3}{2} \sqrt{3}\right)\) (ellipse)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sec \frac{1}{1+x^{2}} $$

Compute the limits in Problems \(53-56 .\) $$ \lim _{h \rightarrow 0} \frac{2^{h}-1}{h} $$

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

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=-|x+5| $$

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\sec x^{2}}{\sec ^{2} x} $$

Find the length of the subtangent to the curve \(y=2^{x}\) at the point \((1,2)\).

Differentiate with respect to the independent variable. $$ f(s)=\frac{4-2 s^{2}}{1-s} $$

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

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=2-|x-3| $$

Lemniscate (a) The curve with equation \(y^{2}=x^{2}-x^{4}\) is shaped like the numeral eight. Find \(\frac{d y^{2}}{d x}\) 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 \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\csc \left(3-x^{2}\right)}{1-x^{2}} $$

Find the length of the subtangent to the curve \(y=\exp \left[x^{2}\right]\) at the point \(\left(2, e^{4}\right)\).

Differentiate with respect to the independent variable. $$ f(s)=\frac{2 s^{3}-4 s^{2}+5 s-7}{\left(s^{2}-3\right)^{2}} $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\log \left(\sqrt[3]{\tan x^{2}}\right) $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=|x+2|-1 $$

Astroid (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 \(-10 \leq x \leq 10\), \(-10 \leq y \leq 10\)

Find the points on the curve \(y=\sin \left(\frac{\pi}{3} x\right)\) that have a horizontal tangent.

Population Growth Suppose that the population size at time \(\underline{t}\) is $$ N(t)=e^{2 t}, \quad t \geq 0 $$ (a) What is the population size at time \(0 ?\) (b) Show that $$ \frac{d N}{d t}=2 N $$

Differentiate with respect to the independent variable. $$ f(x)=\sqrt{x}(x-1) $$

Differentiate the functions with respect to the independent variable. $$ f(u)=\log _{3}\left(3+u^{4}\right) $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{1}{2+x} $$

Kampyle of Eudoxus (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 \(-3 \leq x \leq 3,-10 \leq y \leq 10\)

Find the points on the curve \(y=\cos ^{2} x\) that have a horizontal tangent.

Population Growth Suppose that the population size at time is $$ N(t)=N_{0} e^{r t}, \quad t \geq 0 $$ where \(N_{0}\) is a positive constant and \(r\) is a real number. (a) What is the population size at time 0 ? (b) Show that $$ \frac{d N}{d t}=r N $$