Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).

f'(x) = 10 sin 2x on [-2π, 2π]

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Identify the critical points by setting the derivative f'(x) = 10 sin(2x) equal to zero and solving for x. This gives the equation 10 sin(2x) = 0, which simplifies to sin(2x) = 0.

Solve sin(2x) = 0 for x. The general solution for sin(θ) = 0 is θ = nπ, where n is an integer. Therefore, 2x = nπ, leading to x = nπ/2.

Determine the specific critical points within the interval [-2π, 2π] by substituting integer values for n. Calculate x for n = -4, -3, -2, -1, 0, 1, 2, 3, 4.

Use the first derivative test to classify each critical point as a local maximum, local minimum, or neither. Evaluate the sign of f'(x) on intervals between the critical points to determine where f is increasing or decreasing.

Sketch a possible graph of f by plotting the critical points and indicating the intervals of increase and decrease. Use the behavior of the sine function and the periodicity of f'(x) to guide the shape of the graph.

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Derivatives and Critical Points

The derivative of a function, denoted as f'(x), represents the rate of change of the function at any point. Critical points occur where the derivative is zero or undefined, indicating potential local maxima or minima. To find these points, we set f'(x) = 0 and solve for x, which helps identify where the function's behavior changes.

Increasing and Decreasing Intervals

The intervals on which a function is increasing or decreasing can be determined by analyzing the sign of its derivative. If f'(x) > 0, the function is increasing on that interval; if f'(x) < 0, it is decreasing. This analysis helps in understanding the overall shape and behavior of the function across its domain.

Graph Sketching

Sketching a graph of a function involves plotting its critical points, identifying intervals of increase and decrease, and considering the behavior at the endpoints of the interval. While the exact graph may not be unique, understanding the function's key features allows for a reasonable approximation of its shape, reflecting local maxima, minima, and overall trends.

Functions from derivatives Use the derivative f' to determine the x-coordinates of the local maxima and minima of f, and the intervals on which f is increasing or decreasing. Sketch a possible graph of f (f is not unique).f'(x) = 10 sin 2x on [-2π, 2π]