# Chapter 2 - Differentiation - 2.4 Exercises: 42

(a) Slope$=3$; Number of Complete Cycles=Slope$=3$; Slope=$a=3$. (b)Slope$=\dfrac{1}{2}$ Number of Complete Cycles=Slope$=\dfrac{1}{2}$; Slope $=a=\dfrac{1}{2}$.

#### Work Step by Step

(a)Using the Chain Rule: $\dfrac{d}{dx}\sin{3x}=3\cos{3x}$. $3\cos{0}=3\rightarrow$ Slope of the tangent line is $3$. A complete cycle consists of two humps so graph (a) consists of three complete cycles. $\sin{3x}\rightarrow a=3$. Slope$=3$; Number of complete cycles=Slope$=3$; Slope=$a=3$. (b)Using the Chain Rule: $\dfrac{d}{dx}\sin{\dfrac{1}{2}x}=\dfrac{1}{2}\cos{\dfrac{1}{2}x}$. $\dfrac{1}{2}\cos{0}=\dfrac{1}{2}\rightarrow$ Slope of tangent is $\dfrac{1}{2}$. A complete cycle consists of two humps so graph (b) consists of half a complete cycle. $\sin{\dfrac{1}{2}x}\rightarrow a=\dfrac{1}{2}$. Slope$=\dfrac{1}{2}$ Number of complete cycles=Slope$=\dfrac{1}{2}$; Slope $=a=\dfrac{1}{2}$.

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