Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. The lines tangent to the graph of y=sin x on the interval [−π/2,π/2] have a maximum slope of 1.

Verified step by step guidance

1. First, recall that the slope of the tangent line to the graph of a function y = f(x) at a point x is given by the derivative f'(x). For the function y = sin(x), the derivative is f'(x) = cos(x).
2. Next, consider the interval [−π/2, π/2]. We need to find the maximum value of the derivative f'(x) = cos(x) on this interval, as this will give us the maximum slope of the tangent lines.
3. The function cos(x) is continuous and differentiable on the interval [−π/2, π/2]. To find its maximum value, we can evaluate cos(x) at the critical points and endpoints of the interval.
4. The critical points occur where the derivative of cos(x), which is -sin(x), is zero. This happens when x = 0 within the interval [−π/2, π/2]. Evaluate cos(x) at x = 0, which gives cos(0) = 1.
5. Finally, evaluate cos(x) at the endpoints of the interval: cos(−π/2) = 0 and cos(π/2) = 0. Comparing these values, the maximum value of cos(x) on the interval [−π/2, π/2] is indeed 1, confirming that the maximum slope of the tangent lines is 1.

Key Concepts

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point. For the function y = sin x, the slope of the tangent line is given by its derivative, which is cos x.

Derivatives

The derivative of a function measures how the function's output value changes as its input value changes. For y = sin x, the derivative is cos x, which indicates the slope of the tangent line at any point x. Understanding the behavior of the derivative within a specific interval helps determine maximum and minimum slopes.

Maximum and Minimum Values

In calculus, finding the maximum and minimum values of a function involves analyzing its critical points and endpoints within a given interval. For the function y = sin x on the interval [−π/2, π/2], we can evaluate the derivative to find where it reaches its highest value, which is essential for determining the maximum slope of the tangent lines.