91Ó°ÊÓ

Concept Check Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\) ) $$f(t)=\left(\frac{1}{3}\right)^{1-2 t}$$

Concept Check Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\) ) $$f(t)=\left(\frac{1}{2}\right)^{1-2 t}$$

Use a graphing calculator to solve each equation. Give irrational solutions correct to the nearest hundredth. $$\ln x=-\sqrt[3]{x+3}$$

In calculus, it is shown that $$ e^{x}=1+x+\frac{x^{2}}{2 \cdot 1}+\frac{x^{3}}{3 \cdot 2 \cdot 1}+\frac{x^{4}}{4 \cdot 3 \cdot 2 \cdot 1}+\frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}+\cdots $$ By using more terms, one can obtain a more accurate approximation for \(e^{x}\). Use the terms shown, and replace \(x\) with 1 to approximate \(e^{1}=e\) to three decimal places. Check your result with a calculator.

Explain the error in the following "proof" that \(2<1\). $$\begin{aligned}\frac{1}{9} &<\frac{1}{3} \\\\\left(\frac{1}{3}\right)^{2} &<\frac{1}{3} \\ \log \left(\frac{1}{3}\right)^{2} &<\log \frac{1}{3} \\\2 \log \frac{1}{3} &<1 \log \frac{1}{3} \\ 2 &<1\end{aligned}$$

In calculus, it is shown that $$ e^{x}=1+x+\frac{x^{2}}{2 \cdot 1}+\frac{x^{3}}{3 \cdot 2 \cdot 1}+\frac{x^{4}}{4 \cdot 3 \cdot 2 \cdot 1}+\frac{x^{5}}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}+\cdots $$ By using more terms, one can obtain a more accurate approximation for \(e^{x}\). Use the terms shown, and replace \(x\) with \(-0.05\) to approximate \(e^{-0.05}\) to four decimal places. Check your result with a calculator.

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(f\) has an inverse function \(f^{-1},\) sketch \(f\) and \(f^{-1}\) on the same set of axes.

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(\left.a=10, \text { what is the equation for } y=f^{-1}(x) ? \text { (You need not solve for } y .\right)\)

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(\left.a=e, \text { what is the equation for } y=f^{-1}(x) ? \text { (You need not solve for } y .\right)\)