the point p(3/2,0) lies on the curve y = cos(3πx). a) if q is the point (x,cos(3πx)), use your calculator to…

Explanation:

Step1: Recall slope - formula

The slope $m$ of the secant line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $P(\frac{3}{2},0)$ so $x_1=\frac{3}{2}$ and $y_1 = 0$, and $Q(x,\cos(3\pi x))$ so $x_2=x$ and $y_2=\cos(3\pi x)$. Then the slope formula becomes $m=\frac{\cos(3\pi x)-0}{x - \frac{3}{2}}=\frac{\cos(3\pi x)}{x-\frac{3}{2}}$.

Step2: Calculate slope for $x = 0$

Substitute $x = 0$ into the slope formula: [ \begin{align*} m&=\frac{\cos(3\pi\times0)}{0-\frac{3}{2}}\ &=\frac{\cos(0)}{-\frac{3}{2}}\ &=\frac{1}{-\frac{3}{2}}=-\frac{2}{3}\approx - 0.666667 \end{align*} ]

Step3: Calculate slope for $x = 1.4$

Substitute $x = 1.4$ into the slope formula: [ \begin{align*} m&=\frac{\cos(3\pi\times1.4)}{1.4-\frac{3}{2}}\ &=\frac{\cos(4.2\pi)}{1.4 - 1.5}\ &=\frac{\cos(0.2\pi)}{-0.1}\ &=\frac{\cos(36^{\circ})}{-0.1}\ &\approx\frac{0.809017}{-0.1}=- 8.090170 \end{align*} ]

Step4: Calculate slope for $x = 1.49$

Substitute $x = 1.49$ into the slope formula: [ \begin{align*} m&=\frac{\cos(3\pi\times1.49)}{1.49-\frac{3}{2}}\ &=\frac{\cos(4.47\pi)}{1.49 - 1.5}\ &=\frac{\cos(0.47\pi)}{-0.01}\ &=\frac{\cos(84.6^{\circ})}{-0.01}\ &\approx\frac{0.091760}{-0.01}=-9.176040 \end{align*} ]

Answer:

i) - 0.666667 ii) - 8.090170 iii) - 9.176040