Question

The Derivative of \(\cos \left(x^{2}\right)\) Graph \(y=-2 x \sin \left(x^{2}\right)\) for \(-2 \leq x \leq 3 .\) Then, on screen, graph $$y=\frac{\cos \left[(x+h)^{2}\right]-\cos \left(x^{2}\right)}{h}$$ for \(h=1.0,0.7,\) and \(0.3 .\) Experiment with other values of \(h .\) What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

Short answer

The derivative of \( \cos \left(x^{2}\right) \) is \(-2x \sin(x^{2})\). The graph of the function \(y=\frac{\cos \left[(x+h)^{2}\right]-\cos \left(x^{2}\right)}{h}\) for different values of \(h\) gradually resembles the graph of the derivative as \(h\) approaches zero, thus confirming that the derivative is indeed \(-2x \sin(x^{2})\).

Step-by-step solution

Calculate the derivative using Chain Rule

Start with the formula for the Chain Rule which is \(\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx}\). In the given function \(y=\cos \left( x^{2} \right)\), let \(u = x^{2}\). Therefore \(\frac{du}{dx} = 2x\). Next since \(y = \cos(u)\), the derivative of \(y\) with respect to \(u\) is \(-\sin(u)\). Substituting both derivatives into the Chain Rule gives \(\frac{dy}{dx} = -\sin(u) * 2x = -2x \sin(x^{2})\)

Draw graph of the derivative

With the derivative \(-2x \sin(x^{2})\), this function can be graphed on the interval -2 \(\leq x \leq 3\). The graph will help to visualize the function which will be crucial in further steps.

Compute and graph finite differences

Next you will graph the function \(y=\frac{\cos \left[(x+h)^{2}\right]-\cos \left(x^{2}\right)}{h}\) for \(h=1.0, 0.7,\) and \(0.3\). These equations are formulas for finite differences, which represent approximate derivatives.

Observe what happens as \(h\) approaches 0

By experimenting with decreasing values of \(h\), you should notice that as \(h\) approaches 0, the graph of the finite difference function becomes very similar to -2x \(\sin(x^{2})\). This is because the finite difference formula is an approximation of the derivative, and as \(h\) approaches 0, the difference becomes an increasingly better approximation of the derivative.

Explain the observed behaviour

When \(h \rightarrow 0\), \( \frac{\cos \left[(x+h)^{2}\right] - \cos \left(x^{2}\right)}{h} \) should theoretically converge to the derivative of the function \(-2x \sin(x^{2})\). This converging behaviour observed confirms this hypothesis.

Key concepts

Derivative of Cos(x^2)

Understanding the derivative of a function is crucial in calculus, particularly when the function involves a composition of functions, like \( \cos(x^2) \). To find the derivative, we apply the Chain Rule, an essential tool that breaks down the rate of change of complex functions into simpler parts.

When given \( \cos(x^2) \), we define an inner function \( u = x^2 \), and an outer function \( y = \cos(u) \). Taking the derivative of \( u \) with respect to \( x \) gives us \( 2x \). Deriving \( y \) with respect to \( u \) gives us \( -\sin(u) \). Finally, by multiplying these derivatives, \( \frac{dy}{dx} = -\sin(u) \cdot 2x \), we obtain the derivative \( -2x \sin(x^2) \). This process of applying the Chain Rule helps students grapple with more complex rate-of-change problems.

Graphing Derivatives

Graphing the derivative of a function can significantly enhance our understanding of its behavior. For instance, the derivative of \( \cos(x^2) \), which we found to be \( -2x \sin(x^2) \), provides us with crucial insights when visualized.

A graphical representation helps to solidify concepts such as local maxima, minima, and points of inflection. By graphing \( y = -2x \sin(x^2) \) within the given domain, we can observe where the rate of change is positive, negative, or zero, translating to increases, decreases, or level-offs in the function's value. Students should plot such graphs to better interpret and predict the function's behavior.

Finite Differences Method

The finite differences method is an approximation technique that can give us insight into a function's derivative without needing to calculate it directly. By evaluating the function \( y = \frac{\cos((x+h)^2) - \cos(x^2)}{h} \) for different values of \( h \) and then plotting the results, students learn about the behavior of derivatives in a practical and hands-on manner.

As \( h \) changes, so do the slope calculations, which can be seen as the average rate of change over the interval \( h \) next to \( x \). The closer \( h \) gets to zero, the more these differences resemble the real derivative. This approach allows students to appreciate the derivative as the limit of these finite differences, providing a clear visual and intuitive understanding of the derivative as the instantaneous rate of change at a point.

Limit of a Function as h Approaches 0

The concept of the limit is at the heart of calculus, embodying the idea of approaching a value without necessarily reaching it. When considering the limit as \( h \) approaches zero, we're interested in the behavior of the function \( \frac{\cos((x+h)^2) - \cos(x^2)}{h} \) and how it approximates the derivative of \( \cos(x^2) \) at that point.

As \( h \) becomes smaller, the slope of the secant line between \( x \) and \( x + h \) begins to approach the slope of the tangent line at \( x \)—this is essentially the derivative. In essence, the limit captures the essence of calculus by providing a fundamental way of defining the derivative, helping students understand that the derivative at a point is the result of taking this limit as \( h \) goes toward zero. This convergence is what students will witness when graphing the finite differences for diminishing values of \( h \) and seeing the result approach the graph of the function's derivative.