Chapter 6

Write down a power series \(y=1-x+\cdots\) whose derivative is \(-y\).

Solve staring from \(y_{0}=1\) and from \(y_{0}=-1 .\) Draw both solution on the same graph. $$ \frac{d y}{d t}=2 t $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (2 x) $$

Find the derivatives of the functions $$ 7 e^{7 x} $$

Find these logarithms (or exponents): (a) \(\log _{2} 32\) (b) \(\log _{2}(1 / 32)\) (c) \(\log _{32}(1 / 32)\) (d) \(\log _{32} 2\) (e) \(\log _{10}(10 \sqrt{10})\) (I) \(\log _{2}\left(\log _{2} 16\right)\)ms.

Write down a power series \(y=1+2 x+\cdots\) whose derivative is \(2 y\).

From the definitions of \(\cosh x\) and \(\sinh x\), find their derivatives.

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (2 x+1) $$

Find the derivatives of the functions $$ -7 e^{-7 x} $$

Without a calculator find the values of (a) \(3^{\log , 5}\) (b) \(3^{210065}\) (c) \(\log _{10} 5+\log _{10} 2\) (d) \(\left(\log _{3} b\right)\left(\log _{0} 9\right)\) (e) \(10^{5} 10^{-4} 10^{3}\) (f) \(\log _{2} 56-\log _{2} 7\)

Find two series that are equal to their second derivatives.

Separate, integrate, and solve equations \(1-8 .\) $$ d y / d x=x i y^{2}, \quad y_{0}=1 $$

Solve staring from \(y_{0}=1\) and from \(y_{0}=-1 .\) Draw both solution on the same graph. $$ \frac{d y}{d t}=2 y $$

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

Find the derivatives of the functions $$ \left(e^{x}\right)^{8} $$

Sketch \(y=2^{-x}\) and \(y=\frac{1}{2}\left(4^{x}\right)\) from -1 to 1 on the same graph. Put their mirror images \(x=-\log _{2} y\) and \(x=\log _{4} 2 y\) on a second graph.

By the quotient rule, verify \((\tanh x)^{\prime}=\operatorname{sech}^{2} x\).

Solve staring from \(y_{0}=1\) and from \(y_{0}=-1 .\) Draw both solution on the same graph. $$ \frac{d y}{d t}=-y $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=(\ln x) / x $$

Find the derivatives of the functions $$ \left(e^{-x}\right)^{-8} $$

Derive \(\cosh ^{2} x+\sinh ^{2} x=\cosh 2 x,\) from the definitions.

At \(5 \%\) interest compute the output from \(\$ 1000\) in a year with 6-month and 3-month and weekly compounding.

Separate, integrate, and solve equations \(1-8 .\) $$ d y / d x=(y+1) /(x+1), \quad y_{0}=0 $$

Solve starting from \(y_{0}=10\). At what time does \(y\) increase to 100 or drop to \(1 ?\) $$ \frac{d y}{d t}=4 y $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=x \ln x-x $$

Find the derivatives of the functions $$ 3^{x} $$

Compute without a computer: (a) \(\left.\log _{2} 3+\log _{2}\right\\}\) (b) \(\log _{2}\left(\frac{4}{2}\right)^{10}\) (c) \(\log _{10} 100^{40}\) (d) \(\left(\log _{10} e\right)\left(\log _{\varepsilon} 10\right)\) (e) \(2^{2^{s}} /\left(2^{2}\right)^{3}\) (f) \(\log _{\alpha}(1 / e)\)

With the quick method \(\ln (1+x) \approx x,\) estimate \(\ln (1-1 / \pi)^{n}\)and \(\ln (1+2 / n)^{n}\). Then take exponentials to find the two limits.

Solve starting from \(y_{0}=10\). At what time does \(y\) increase to 100 or drop to \(1 ?\) $$ \frac{d y}{d t}=4 t $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\log _{10} x $$

Find the derivatives of the functions $$ e^{x \ln 3} $$

Solve the following equations for \(x\) : (a) \(\log _{10}\left(10^{x}\right)=7\) (b) \(\log 4 x-\log 4=\log 3\) (c) \(\log _{x} 10=2\) (d) \(\log _{2}(1 / x)=2\) (e) \(\log x+\log x=\log 8\) (f) \(\log _{x}\left(x^{x}\right)=5\)

With the slow method multiply out the three terms of \(\left(1-\frac{1}{2}\right)^{2}\) and the five terms of \(\left(1-\frac{1}{4}\right)^{4}\). What are the first three terms of \((1-1 / n)^{n},\) and what are their limits as \(n \rightarrow \infty ?\).

Solve starting from \(y_{0}=10\). At what time does \(y\) increase to 100 or drop to \(1 ?\) $$ \frac{d y}{d t}=e^{4 t} $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (\sin x) $$

Find the derivatives of the functions $$ (2 / 3)^{x} $$

The logarithm of \(y=x^{n}\) is \(\log y=\) ______ .

Prove \(\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y\) by changing to cxponentials. Then the \(x\) -derivative gives \(\cosh (x+y)=\)

Separate, integrate, and solve equations \(1-8 .\) $$ d y / d t=e^{t-y}, \quad y_{0}=e $$

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (\ln x) $$

Find the derivatives of the functions $$ 4^{4 x} $$

Find the derivatives of the functions \(9-18:\) \(\cosh (3 x+1)\)

Knowing that \((1+1 / n)^{n} \rightarrow e_{1}\) explain \((1+1 / n)^{2 n} \rightarrow e^{2}\) and \((1+2 / N)^{N} \rightarrow e^{2}\).

Suppose the rate of growth is proportional to \(\sqrt{y}\) instead of \(y .\) Solve \(d y i d t=c \sqrt{y}\) starting from \(y_{0}\)

Draw a field of "tangent arrows" for \(y^{\prime}=-y,\) with the solution curves \(y=e^{-x}\) and \(y=-e^{-x}\)

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

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

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

What are the limits of \(\left(1+1 / n^{2}\right)^{n}\) and \((1+1 / n)^{n^{2}} ?\) OK to use a calculator to guess these limits.

Draw a direction field of arrows for \(y^{\prime}=y-1,\) with solution curves \(y=e^{x}+1\) and \(y=1\).