Chapter 1

Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic $y=f(x)=x^3+$ ax. Find the inverse of the following cubics using the substitution (known as Vieta's substitution) $x=z-a/(3z).$ Be sure to determine where the function is one-to-one. $$f(x)=x^3+2x$$

Amplitude and period Identify the amplitude and period of the following functions. $$g(\theta )=3\cos (\theta /3)$$

Simplify the difference quotients $\frac{f(x+h)-f(x)}{h}$ and $\frac{f(x)-f(a)}{x-a}$ by rationalizing the numerator. $$f(x)=\sqrt{x}$$

Finding the inverse of a cubic polynomial is equivalent to solving a cubic equation. A special case that is simpler than the general case is the cubic $y=f(x)=x^3+$ ax. Find the inverse of the following cubics using the substitution (known as Vieta's substitution) $x=z-a/(3z).$ Be sure to determine where the function is one-to-one. $$f(x)=x^3+4x-1$$

Amplitude and period Identify the amplitude and period of the following functions. $$p(t)=2.5\sin (\frac{1}{2}(t-3))$$

Simplify the difference quotients $\frac{f(x+h)-f(x)}{h}$ and $\frac{f(x)-f(a)}{x-a}$ by rationalizing the numerator. $$f(x)=\sqrt{1-2x}$$

Prove that $({\log }_{b}c)({\log }_{c}b)=1,$ for $b>0$ $c>0,b\ne 1,$ and $c\ne 1$

Graphing sine and cosine functions Beginning with the graphs of $y=\sin x$ or $y=\cos x,$ use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. $$g(x)=-2\cos (x/3)$$

Graphing sine and cosine functions Beginning with the graphs of $y=\sin x$ or $y=\cos x,$ use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. $$p(x)=3\sin (2x-\pi /3)+1$$

A cylindrical tank with a cross-sectional area of $100{\mathrm{cm}}^2$ is filled to a depth of $100\mathrm{cm}$ with water. At $t=0,$ a drain in the bottom of the tank with an area of $10{\mathrm{cm}}^2$ is opened, allowing water to flow out of the tank. The depth of water in the tank at time $t\ge 0$ is $d(t)=(10-2.2t{)}^2$. a. Check that $d(0)=100,$ as specified. b. At what time is the tank empty? c. What is an appropriate domain for $d?$