Let 𝑓 be the function given by 𝑓(𝑥)=4cos𝑥. As shown below, the graph of 𝑓 crosses the y-axis at point P (0,4) and the x-axis at point Q (𝜋2,0).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.
Question
Let 𝑓 be the function given by 𝑓(𝑥)=4cos𝑥. As shown below, the graph of 𝑓 crosses the y-axis at point P (0,4) and the x-axis at point Q (𝜋2,0).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.
Solution
The line PQ has a slope of -4/(π/2) = -8/π.
The derivative of f(x) = 4cos(x) is f'(x) = -4sin(x).
We want to find the x-coordinate where the slope of the tangent line (given by f'(x)) is equal to the slope of line PQ. So we set -4sin(x) = -8/π and solve for x.
4sin(x) = 8/π sin(x) = 2/π
The x-coordinate where sin(x) = 2/π and x is between 0 and π/2 is x = arcsin(2/π).
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