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Magazine Problem 88
Identify the amplitude and period of the following functions. $$f(\theta)=2 \sin 2 \theta$$
Problem 89
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 \(y=f(x)=x^{3}+a x\). Find the inverse of the following cubics using the substitution (known as Vieta's substitution) \(x=z-a /(3 z) .\) Be sure to determine where the function is one-to-one. $$f(x)=x^{3}+2 x$$
Problem 89
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{x}$$
Problem 89
Identify the amplitude and period of the following functions. $$g(\theta)=3 \cos (\theta / 3)$$
Problem 90
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{1-2 x}$$
Problem 90
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 \(y=f(x)=x^{3}+a x\). Find the inverse of the following cubics using the substitution (known as Vieta's substitution) \(x=z-a /(3 z) .\) Be sure to determine where the function is one-to-one. $$f(x)=x^{3}-2 x$$
Problem 90
Identify the amplitude and period of the following functions. $$p(t)=2.5 \sin \left(\frac{1}{2}(t-3)\right)$$
Problem 91
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=-\frac{3}{\sqrt{x}}$$
Problem 91
Identify the amplitude and period of the following functions. $$q(x)=3.6 \cos (\pi x / 24)$$
Problem 92
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{x^{2}+1}$$
