Answer
(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 $=a=\dfrac{1}{2}$.
