Graph the function y=2 cos2 x for -2 ≤x ≤3.5 . Then, on the same screen, graph y=(sin2(x+h)-sin2

step 1 For the following problem, we're going to consider the function F of X plus H. We're going to do

Graph the function y=2 cos2 x for -2 ≤x ≤3.5 . Then, on the...

Graph the function $y=2 \cos 2 x$ for $-2 \leq x \leq 3.5 .$ Then, on the same screen, graph $$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for $h=1.0,0.5,$ and $0.2 .$ Experiment with other values of $h$ including negative values. What do you see happening as $h \rightarrow 0 ?$ Explain this behavior.

Graph the function $y=2 \cos 2 x$ for $-2 \leq x \leq 3.5 .$ Then, on the same screen, graph
$$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$
for $h=1.0,0.5,$ and $0.2 .$ Experiment with other values of $h$ including negative values. What do you see happening as $h \rightarrow 0 ?$ Explain this behavior.

Key Concepts

Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as h approaches zero. It represents the instantaneous rate of change or the slope of the tangent line to the function at that point. In this problem, the derivative of sin(2x) is found to be 2cos(2x), which is confirmed graphically by showing that as h approaches 0, the graphs of the difference quotient converge to the graph of 2cos(2x).

Limit Process

The limit process involves examining what happens to an expression as a parameter—in this case, h—approaches a particular value (zero). Here, taking the limit of the difference quotient as h approaches zero provides the exact derivative of the function. This showcases how approximate average rates of change can be refined into an exact instantaneous rate of change when h is infinitesimally small.

Difference Quotient

The difference quotient is an expression that measures the average rate of change of a function over an interval. It is given by [f(x+h) - f(x)]/h and serves as the foundational concept for defining the derivative of a function. In this context, it is used to approximate the slope of the function sin(2x) for small values of h and observe how that approximation improves as h tends toward zero.

Trigonometric Derivatives

Trigonometric derivatives refer to the derivatives of trigonometric functions. A key example is the derivative of sin(2x), which, by the chain rule, is 2cos(2x). Understanding these derivatives is essential when working with oscillatory functions and graphing their behavior, as demonstrated in this problem where the difference quotient approximates the derivative of sin(2x) for various values of h.

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