Chapter 3

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=e^{-4 t}\) 14\. \(P=e^{-0.2 t}\) 15\. \(P=50 e^{-0.6 t}\) 16\. \(P=200 e^{0.12 t}\) 17\. \(P(t)=3000(1.02)^{t}\) 18\. \(P(t)=12.41(0.94)^{t}\) 19\. \(P(t)=C e^{t}\). 20\. \(y=B+A e^{t}\) 21\. \(f(x)=A e^{x}-B x^{2}+C\) 22\. \(y=10^{x}+\frac{10}{x}\) 23\. \(R=3 \ln q\) 24\. \(D=10-\ln p\) 25\. \(y=t^{2}+5 \ln t\) 26\. \(R(q)=q^{2}-2 \ln q\) 27\. \(y=x^{2}+4 x-3 \ln x\) 28\. \(f(t)=A e^{t}+B \ln t\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 x^{2}+7 x-9$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ y=6 \sin (2 t)+\cos (4 t) $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=\frac{x^{2}+3}{x} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P=e^{-0.2 t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=8 t^{3}-4 t^{2}+12 t-3$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(x)=x^{2} \cos x $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=t e^{-t^{2}} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P=50 e^{-0.6 t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=4.2 q^{2}-0.5 q+11.27$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(x)=2 x \sin (3 x) $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(z)=\sqrt{z} e^{-z} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P=200 e^{0.12 t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=-3 x^{4}-4 x^{3}-6 x+2$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(\theta)=\theta^{3} \cos \theta $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P(t)=3000(1.02)^{t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$g(t)=\frac{1}{t^{5}}$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ g(p)=p \ln (2 p+1) $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(t)=t e^{5-2 t} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P(t)=12.41(0.94)^{t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(z)=-\frac{1}{z^{6.1}}$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(t)=\frac{t^{2}}{\cos t} $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(w)=\left(5 w^{2}+3\right) e^{w^{2}} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(P(t)=C e^{t}\).

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=\frac{1}{r^{7 / 2}}$$

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ f(\theta)=\frac{\sin \theta}{\theta} $$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=x \cdot 2^{x} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=B+A e^{t}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=\sqrt{x}$$

Find the equation of the tangent line to the graph of \(y=\sin x\) at \(x=\pi\). Graph the function and the tangent line on the same axes.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ w=\left(t^{3}+5 t\right)\left(t^{2}-7 t+2\right) $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(f(x)=A e^{x}-B x^{2}+C\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(\theta)=\frac{1}{\sqrt[3]{\theta}}$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ z=\left(t e^{3 t}+e^{5 t}\right)^{9} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=10^{x}+\frac{10}{x}\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=\sqrt{\frac{1}{x^{3}}}$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=\frac{x}{e^{x}} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(R=3 \ln q\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 t^{5}-5 \sqrt{t}+\frac{7}{t}$$

Find equations of the tangent lines to the graph of \(f(x)=\) \(\sin x\) at \(x=0\) and at \(x=\pi / 3\). Use each tangent line to approximate \(\sin (\pi / 6)\). Would you expect these results to be equally accurate, since they are taken equally far away from \(x=\pi / 6\) but on opposite sides? If the accuracy is different, can you account for the difference?

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ w=\frac{3 z}{1+2 z} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(D=10-\ln p\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=z^{2}+\frac{1}{2 z}$$

A company's monthly sales, \(S(t)\), are seasonal and given as a function of time, \(t\), in months, by $$ S(t)=2000+600 \sin \left(\frac{\pi}{6} t\right) . $$ (a) Graph \(S(t)\) for \(t=0\) to \(t=12\). What is the maximum monthly sales? What is the minimum monthly sales? If \(t=0\) is January 1 , when during the year are sales highest? (b) Find \(S(2)\) and \(S^{\prime}(2)\). Interpret in terms of sales.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ z=\frac{1-t}{1+t} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=t^{2}+5 \ln t\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 t^{2}+\frac{12}{\sqrt{t}}-\frac{1}{t^{2}}$$

A boat at anchor is bobbing up and down in the sea. The vertical distance, \(y\), in feet, between the sea floor and the boat is given as a function of time, \(t\), in minutes, by $$ y=15+\sin (2 \pi t) $$ (a) Find the vertical velocity, \(v\), of the boat at time \(t\). (b) Make rough sketches of \(y\) and \(v\) against \(t\).

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=\frac{e^{x}}{1+e^{x}} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(R(q)=q^{2}-2 \ln q\)