Chapter 4, Problem 76RE

To determine

To write: a formula that gives the maximum possible slope for a sinusoid with amplitude A and period p. justify your answer.

Answer to Problem 76RE

The formula that gives the maximum possible slope, is ${\frac{dy}{dx}}_{\max }=AB.$

  $\text{Where}A\text{is}\text{the}\text{amplitude}\text{and}B=\frac{2\pi }{p}.$

Explanation of Solution

Given information: The general equation of a sinusoid is $y=A\sin (Bx+C)+D,$ where

only the constants A and B affect the amplitude and period.

Calculation:

The general equation of a sinusoid is $y=A\sin (Bx+C)+D,$

Slope of this equation is $\frac{dy}{dx}$ .

  $\begin{array}{l}y=A\sin (Bx+C)+D.\\ \frac{dy}{dx}=\frac{d(A\sin (Bx+C)+D)}{dx}=AB\cos (Bx+C).\\ \text{The}\text{maximum}\text{of}\cos (Bx+C)=1.\\ So,\\ {\frac{dy}{dx}}_{\max }=\max (AB\cos (Bx+C))=AB\max (\cos (Bx+C))=AB\times 1=AB.\\ {\frac{dy}{dx}}_{\max }=AB.\\ \text{Where}A\text{is}\text{the}\text{amplitude}\text{and}B=\frac{2\pi }{p}.\\ \text{Justification}\text{.}\\ \text{When}\text{A=2}\text{}\text{and}\text{p=}\pi \text{then}B=2\\ {\frac{dy}{dx}}_{\max }=3\times 2=6.\text{is}\text{the}\text{maximum}\text{slope}\text{of}\text{sin}\text{function}.\end{array}$