Problem 1
What are the values of the following logarithms? \((a) \log _{10} 10,000\) \((c) \log _{3} 81\) (b) \(\log _{10} 0.0001\) \((d) \log _{5} 3,125\)
Problem 1
Find the instantaneous rate of growth: (a) \(y=5 t^{2}\) (b) \(y=a t^{c}\) (c) \(y=a b^{t}\) \((d) y=2^{t}\left(t^{2}\right)\) (e) \(y=t / 3^{t}\)
Problem 2
Plot in a single diagram the graphs of the exponential functions \(y=4^{t}\) and \(y=3\left(4^{t}\right)\) (a) Do the two graphs display the general positional relationship suggested in Fig. \(10.2 b ?\) (b) Do the two curves have the same \(y\) intercept? Why? (c) Sketch the graph of the function \(y=\frac{3}{2}\left(4^{t}\right)\) in the same diagram.
Problem 2
Evaluate the following: \((a) \ln e^{7}\) (c) \(\ln \left(1 / e^{3}\right)\) \((e)\left(e^{\lambda \cdot 3}\right) !\) \((b) \log _{e} e^{-4}\) \((d) \log _{e}\left(1 / e^{2}\right)\) (f) \(\ln e^{x}-e^{\ln x}\)
Problem 2
Given the function \(\phi(x)=e^{2 x}:\) (a) Write the polynomial part \(P_{n}\) of its Maclaurin series. (b) Write the Lagrange form of the remainder \(R_{n}\). Determine whether \(R_{n} \rightarrow 0\) as \(n \rightarrow \infty,\) that is, whether the series is convergent to \(\phi(x)\) (c) If convergent, so that \(\phi(x)\) may be expressed as an infinite series, write out this series.
Problem 2
(a) Sketch a graph of the exponential function \(y=A e^{r t} ;\) indicate the value of the vertical intercept. (b) Then sketch the graph of the log function \(t=\frac{\ln y-\ln A}{r},\) and indicate the value of the horizontal intercept.
Problem 2
If population grows according to the function \(H=H_{0}(2)^{b t}\) and consumption by the function \(C=C_{0} e^{o t},\) find the rates of growth of population, of consumption, and of per capita consumption by using the natural log.
Problem 3
Find the inverse function of \(y=\sigma b^{c t}\)
Problem 3
Taking for granted that \(e^{t}\) is its own derivative, use the chain rule to find \(d y\) jd for the following: (a) \(y=e^{5 t}\) (b) \(y=4 e^{3 t}\) (c) \(y=6 e^{-2 t}\)
Problem 3
Find the derivatives of: $$\text { (a) } y=\ln \left(7 t^{5}\right)$$ $$\text { (b) } y=\ln \left(a t^{c}\right)$$ $$\text { (c) } y=\ln (t+19)$$ $$\text { (d) } y=5 \ln (t+1)^{2}$$ $$\text { (e) } y=\ln x-\ln (1+x)$$ $$\text { (f) } y=\ln \left[x(1-x)^{8}\right]$$ $$\text { (g) } y=\ln \left(\frac{2 x}{1+x}\right)$$ $$\text { (h) } y=5 x^{4} \ln x^{2}$$
