Chapter 6

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln \left((4 x)^{7}\right) $$

Find the derivatives of the functions $$ e^{1 /(1+x)} $$

The number \(2^{1000}\) has approximately how many (decimal) digits?

Find the derivatives of the functions \(9-18:\) \(1 / \cosh x\)

Find the indefinite (or definite) integral in \(11-24\). $$ \int \frac{d t}{3 t} $$

Find the derivatives of the functions $$ e^{\ln x}+x^{\ln e} $$

Find the derivatives of the functions \(9-18:\) \(\sinh (\ln x)\)

Find the indefinite (or definite) integral. $$ \int \frac{d x}{1+x} $$

Find the derivatives of the functions $$ x e^{1 / x} $$

Find the derivatives of the functions \(9-18:\) \(\cosh ^{2} x+\sinh ^{2} x\)

Take three steps of \(y(t+1)=2 y(t)\) from \(y_{0}=1\).

Find the indefinite (or definite) integral. $$ \int_{0}^{1} \frac{d x}{3+x} $$

Find the derivatives of the functions $$ x e^{x}-e^{x} $$

Find the derivatives of the functions \(9-18:\) \(\cosh ^{2} x-\sinh ^{2} x\)

Questions \(11-19\) are about the graphs of \(y=b^{x}\) and \(x=\log _{8} y .\)Draw semilog graphs of \(y=10^{1-x}\) and \(y=1(\sqrt{10})\).

Take three steps of \(y(t+1)=2 y(t)+1\) from \(y_{0}=0\).

Find the indefinite (or definite) integral. $$ \int_{0}^{1} \frac{d t}{3+2 t} $$

Find the derivatives of the functions $$ x^{2} e^{x}-2 x e^{x}+2 e^{x} $$

Solve the difference equations $$ y(t+1)=3 y(t), y_{0}=4 $$

The population of Cairo grew from 5 million to 10 million in 20 years. From \(y^{\prime}=c y\) find \(c .\) When was \(y=8\) million?

Find the indefinite (or definite) integral. $$ \int_{0}^{2} \frac{x d x}{x^{2}+1} $$

Find the derivatives of the functions $$ \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} $$

Find the derivatives of the functions \(9-18:\) \((1+\tanh x) /(1-\tanh x)\)

Solve the difference equations $$ y(t+1)=\frac{1}{4} y(t), y_{0}=1 $$

The populations of New York and Los Angeles are growing at \(1 \%\) and \(1.4 \%\) a year. Starting from 8 million (NY) and 6 million (LA), when will they be equal?

Find the indefinite (or definite) integral. $$ \int_{0}^{2} \frac{x^{3} d x}{x^{2}+1} $$

Find the derivatives of the functions $$ e^{\ln \left(x^{2}\right)}+\ln \left(e^{x^{2}}\right) $$

Find the derivatives of the functions \(9-18:\) \(\sinh ^{6} x\)

Questions \(11-19\) are about the graphs of \(y=b^{x}\) and \(x=\log _{8} y .\) Draw your own semilog paper and plot the data $$ y=7,11,16,28,44 \text { at } x=0,1 / 2,1,3 / 2,2 $$

Suppose the value of \(\$ 1\) in Japanese yen decreases at \(2 \%\) per year. Starting from \(\$ 1=Y 240,\) when will 1 dollar equal 1 yen?

Find the derivatives of the functions $$ e^{\sin x}+\sin e^{x} $$

Find the derivatives of the functions \(9-18:\) \(\ln (\operatorname{sech} x+\tanh x)\)

Solve the difference equations $$ y(t+1)=y(t)-1, y_{0}=0 $$

Find the indefinite (or definite) integral. $$ \int_{2}^{e} \frac{d x}{x(\ln x)^{2}} $$

Find the derivatives of the functions $$ x^{-1 / x} \text { (which is } e-\text { ) } $$

Find the derivatives of the functions \(9-18:\) Find the minimum value of \(\cosh (\ln x)\) for \(x>0\)

Solve the difference equations $$ y(t+1)=3 y(t)+1, y_{0}=0 $$

Problems \(13-26\) deal with logistic equations \(y^{\prime}=c y-b y^{2}\). Suppose Pittsburgh grows from \(y_{0}=1\) million people in 1900 to \(y=3\) million in the year 2000 . If the growth rate is \(y^{\prime}=12,000 /\) year in 1900 and \(y^{\prime}=30,000 /\) year in \(2000,\) substitute in the logistic equation to find \(c\) and \(b\). What is the steady state? Extra credit: When does \(y=y_{s} / 2=c / 2 b ?\)

If \(y=1000\) at \(t=3\) and \(y=3000\) at \(t=4\) (exponential growth), what was \(y_{0}\) at \(t=0\) ?

Find the indefinite (or definite) integral. $$ \int \frac{\cos x d x}{\sin x} $$

The difference between \(e\) and \((1+1 / n)^{n}\) is approximately \(\mathrm{Ce} / n .\) Subtract the calculated values for \(n=10,100,1000\) from 2.7183 to discover the number \(C\).

Find the derivatives of the functions \(9-18:\) From \(\tanh x=\frac{3}{5}\) find \(\operatorname{sech} x, \cosh x, \sinh x, \operatorname{coth} x, \operatorname{csch} x\)

Problems \(13-26\) deal with logistic equations \(y^{\prime}=c y-b y^{2}\). Suppose \(c=1\) but \(b=-1\), giving cooperation \(y^{\prime}=y+y^{2}\) Solve for \(\mu(t)\) if \(y_{0}=1 .\) When does \(y\) become infinite?

If \(y=100\) at \(t=4\) and \(y=10\) at \(t=8\) (exponential decay) when will \(y=1\) ? What was \(y_{0}\) ?

Find the indefinite (or definite) integral. $$ \int_{0}^{\pi / 4} \tan x d x $$

By algebra or a calculator find the limits of \((1+1 / n)^{2 n}\) and \((1+1 / n)^{\sqrt{n}}\).

Questions \(20-29\) are about the derivative \(d y / d x=c b^{x}\). If the slope of \(\log x\) is \(1 / c x,\) find the slopes of \(\log (2 x)\) and \(\log \left(x^{2}\right)\) and \(\log \left(2^{2}\right)\)

Find the indefinite (or definite) integral. $$ \int \tan 3 x d x $$

Find the indefinite (or definite) integral. $$ \int \cot 3 x d x $$

Compare the number of correct decimals of \(e\) for \((1.001)^{1000}\) and \((1.0001)^{10000}\) and if possible \((1.00001)^{100000}\). Which power \(n\) would give all the decimals in \(2.71828 ?\)