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Chapter 3: Problem 92

Graph \(y=-2 x \sin \left(x^{2}\right)\) for \(-2 \leq\) \(x \leq 3 .\) Then, on the same 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

Expert verified
As \( h \rightarrow 0 \), the second function approximates the derivative of \( \cos(x^2) \).

Step by step solution

01

Understanding the Functions

We need to graph two functions: \( y = -2x \sin(x^2) \) and \( y = \frac{\cos((x+h)^2) - \cos(x^2)}{h} \). The goal is to observe how the second function behaves for different values of \( h \) as \( h \) approaches zero.
02

Start with the First Function

We graph \( y = -2x \sin(x^2) \) over the interval \(-2 \leq x \leq 3\). This function has oscillations due to the sine component, and the \(-2x\) factor affects its amplitude and direction.
03

Graph the Second Function for h = 1.0

Now graph \( y = \frac{\cos((x+1.0)^2) - \cos(x^2)}{1.0} \). This is the symmetric difference quotient for the function \( \cos(x^2) \), a basic finite difference approximation for derivatives.
04

Graph the Second Function for h = 0.7

Graph \( y = \frac{\cos((x+0.7)^2) - \cos(x^2)}{0.7} \). This represents a finer approximation of the derivative as \( h \) decreases.
05

Graph the Second Function for h = 0.3

Proceed by graphing \( y = \frac{\cos((x+0.3)^2) - \cos(x^2)}{0.3} \). The function should start resembling the derivative of \( \cos(x^2) \) more closely now.
06

Experiment with Smaller Values of h

Try other values of \( h \), even smaller than 0.3, such as 0.1 or 0.01. Observe that as \( h \rightarrow 0 \), the graph approximates the derivative of \( \cos(x^2) \).
07

Conclusion - Behavior as h Approaches 0

As \( h \) approaches zero, the difference quotient approximates the derivative of the function more accurately. This behavior is consistent with the notion that the difference quotient geerally represents the derivative as \( h \rightarrow 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference Quotient
The difference quotient is a mathematical expression that helps approximate the derivative of a function. It looks like this for a function \( f(x) \): \[ \frac{f(x+h) - f(x)}{h} \] This formula calculates the average rate of change of the function \( f(x) \) over the interval from \( x \) to \( x+h \). Here's a simple breakdown:
  • The numerator \( f(x+h) - f(x) \) shows the change in the function's output over the interval.
  • The denominator \( h \) represents the change in the input over the same interval.
As \( h \) gets smaller, the quotient becomes an increasingly accurate estimate of the function's derivative at point \( x \). This is why in calculus, the limit of the difference quotient as \( h \rightarrow 0 \) gives us the exact derivative.
Derivatives
Derivatives are central concepts in calculus used to describe how a function changes as its input changes. Think of a derivative as the slope of a curve at any particular point. When we talk about derivatives:
  • They tell us the rate of change of the function value with respect to the change in input.
  • Visually, this corresponds to the slope of the tangent line to the function's graph at a given point.
In the context of our problem, the derivative of \( \cos(x^2) \) is estimated using the difference quotient with decreasing values of \( h \). As \( h \) approaches zero, the calculated slopes become closer to the actual derivative.
Sine and Cosine Functions
Sine and cosine functions are fundamental in trigonometry. They are periodic functions, meaning they repeat their values in regular intervals. These functions are crucial for modeling oscillations:
  • The sine function, \( \sin(x) \), oscillates between -1 and 1. It starts at 0, peaks at \( \pi/2 \), and returns to 0 at \( \pi \).
  • The cosine function, \( \cos(x) \), also oscillates between -1 and 1 but starts at 1, drops to 0 at \( \pi/2 \), and returns to -1 at \( \pi \).
  • Both functions are used to describe wave-like phenomena and vibrations, such as sound waves.
In our problem, \( \cos(x^2) \) implies the cosine function being applied to \( x^2 \), affecting its oscillation pattern.
Function Approximation
Function approximation is about estimating a complex function using simpler functions or expressions. This helps to analyze and predict function behavior with manageable complexity. In calculus:
  • The difference quotient provides a method to approximate the derivative of a function.
  • For small \( h \), the quotient reflects how the function behaves over tiny increments, thus closely approximating the derivative.
  • Graphing different approximations as \( h \) decreases lets us visually grasp the function smoothening into the derivative.
By graphing these approximations, we see how the estimates improve, and the transition becomes quite evident as \( h \rightarrow 0 \), showcasing the elegance and utility of calculus.

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