1) let ( f ) be the function given by ( f(x)=3cos(x) ). as shown above, the graph of ( f ) crosses the ( y…

1) let ( f ) be the function given by ( f(x)=3cos(x) ). as shown above, the graph of ( f ) crosses the ( y )-axis at point ( p ) and the ( x )-axis at point ( q ).\n a). write an equation for the line passing through the points ( p ) and ( q ).\n b) find the ( x )-coordinate of the point on the graph of ( f ), between points ( p ) and ( q ), at which the line tangent to the graph of ( f ) is parallel to line ( pq ).

1) let ( f ) be the function given by ( f(x)=3cos(x) ). as shown above, the graph of ( f ) crosses the ( y )-axis at point ( p ) and the ( x )-axis at point ( q ).\n a). write an equation for the line passing through the points ( p ) and ( q ).\n b) find the ( x )-coordinate of the point on the graph of ( f ), between points ( p ) and ( q ), at which the line tangent to the graph of ( f ) is parallel to line ( pq ).

Answer

Explanation:

Step1: Find the slope of line (PQ)

The formula for the slope (m) of a line passing through two points ((x_1,y_1)) and ((x_2,y_2)) is (m=\frac{y_2 - y_1}{x_2 - x_1}). Here, (P(0,3)) and (Q(\frac{\pi}{2},0)), so (m=\frac{0 - 3}{\frac{\pi}{2}-0}=-\frac{6}{\pi}).

Step2: Use the point - slope form of a line

The point - slope form of a line is (y - y_1=m(x - x_1)). Using the point (P(0,3)) and (m =-\frac{6}{\pi}), we substitute (x_1 = 0,y_1 = 3) into the formula. (y-3=-\frac{6}{\pi}(x - 0)), which simplifies to (y=-\frac{6}{\pi}x + 3).

Answer:

The equation of the line (PQ) is (y =-\frac{6}{\pi}x+3).