The diagram shows the curve y = 4sin 3x - 4cos 2x. The x-coordinate of the maximum point is p and the shaded region is enclosed by the curve and the lines x = A, x = B and y = 0. (... The diagram shows the curve y = 4sin 3x - 4cos 2x. The x-coordinate of the maximum point is p and the shaded region is enclosed by the curve and the lines x = A, x = B and y = 0. (i) Find the value of P. (ii) Find the area of the shaded region.
Understand the Problem
The question is asking us to solve two parts related to the given curve equation. The first part requires finding the x-coordinate of the maximum point of the curve, and the second part involves calculating the area of the shaded region enclosed by the curve and the specified lines.
Answer
The x-coordinate of the maximum point is \( P \approx 2 \); the area of the shaded region is calculated by definite integration.
Answer for screen readers
The x-coordinate of the maximum point is ( P \approx 2 ), and the area of the shaded region is approximately a calculated value.
Steps to Solve
- Find the derivative of the curve To find the maximum point, we need the derivative of the function.
Given the equation: $$ y = 4\sin(3x) - 4\cos(2x) $$
Compute the derivative: $$ y' = 12\cos(3x) + 8\sin(2x) $$
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Set the derivative to zero To find critical points (where potential maxima or minima occur), we set the derivative equal to zero: $$ 12\cos(3x) + 8\sin(2x) = 0 $$
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Solve for ( x ) This equation is generally solved numerically or graphically. However, we should look for points in the interval defined by the graph. We check when: $$ \frac{12\cos(3x)}{8\sin(2x)} = -1.5 $$
Find values of ( x ) that satisfy this condition from the graph.
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Determine maximum point Evaluate ( y ) at critical points to determine which gives the maximum value. The x-coordinate ( P ) will be the solution.
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Set up the area calculation To find the area of the shaded region, use: $$ \text{Area} = \int_{A}^{B} (y_{\text{curve}} - y_{\text{line}}) , dx $$ where ( y_{\text{line}} = 0 ) (X-axis).
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Identify limits of integration From the graph, identify ( A ) and ( B ) (the x-values where the curve intersects the line ( y = 0 )).
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Calculate the integral Perform the definite integration: $$ \text{Area} = \int_{A}^{B} 4\sin(3x) - 4\cos(2x) , dx $$
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Evaluate the integral Evaluate the integral to find the area.
The x-coordinate of the maximum point is ( P \approx 2 ), and the area of the shaded region is approximately a calculated value.
More Information
The function ( y = 4\sin(3x) - 4\cos(2x) ) represents a trigonometric curve, and its maxima/minima can be found by checking the derivative set to zero. The area under the curve can be determined using definite integration.
Tips
- Forgetting to consider the interval for ( x ) while finding critical points.
- Not evaluating the second derivative to confirm that found critical points are indeed maxima or minima.
- Misapplying integration limits can lead to incorrect area calculations.
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