Problem 17 - Calculus Early Transcendentals

Question

a. Write the equation of the line that represents the linear approximation to the following functions at the given point a. b. Graph the function and the linear approximation at a. c. Use the linear approximation to estimate the given function value. d. Compute the percent error in your approximation, \(100 |\) approximation \(-\) exact \(|/|\) exact \(|,\) where the exact value is given by a calculator. \(f(x)=\cos x ; a=0 ; f(-0.01)\)

Step-by-Step Solution

Answer

Answer: The percent error of the linear approximation is approximately 0.005%.

Step 1: Find the equation of the linear approximation

To find the equation of the linear approximation at a=0, we need to find the tangent line to the function at this point. For this, we need two pieces of information: the point (x, f(x)) and the slope of the function at x = 0. First calculate the function value at a=0: \(f(0)=\cos(0)=1\) Now, we need to find the slope of the function at a=0 by finding the first derivative of the function and calculating its value at this point: \(f'(x) = -\sin(x)\) \(f'(0)=-\sin(0)=0\) Now that we have both the point (0, 1) and the slope 0, we can find the equation of the linear approximation: \(y=f(a)+f'(a)(x-a) \Rightarrow y=1+0(x-0) \Rightarrow y=1\)

Step 2: Graph the function and the linear approximation

To graph both the function and the linear approximation, you need to plot the curves of \(f(x) = \cos x\) and the linear approximation \(y=1\). You can use your favorite graphing software or graphing calculator to plot these and see the similarities and differences between the two curves.

Step 3: Estimate the value of the function at x = -0.01

Using the linear approximation equation from step 1, we can estimate the value of the function at x = -0.01: \(y=1+0(-0.01-0) = 1\) So, our linear approximation estimate is 1.

Step 4: Compute the percent error

Now that we have our approximation, we need to compute the percent error. To do this, first find the exact value of the function (using a calculator) at x=-0.01: \(exact = \cos(-0.01) \approx 0.99995\) Now, compute the percent error using the formula: \(percent \ error = \frac{100 |approximation \ - \ exact|}{|exact|} = \frac{100 |1 - 0.99995|}{0.99995} \approx 0.005 \%\) The percent error in our approximation is approximately 0.005%.

Key Concepts

Tangent Line

In calculus, a tangent line to a function at a given point provides an excellent way to approximate the function near that point. Think of it as the best straight-line approximation of the curve's behavior. The tangent line touches the curve only at one specific point without cutting across it.

• The tangent line equation can be found using the formula: \[y = f(a) + f'(a)(x-a)\]. In this equation:
  • \(f(a)\) is the function value at the point \(a\).
  • \(f'(a)\) is the derivative (slope) of the function at \(a\).
• The tangent line's purpose in linear approximation is to provide a close estimate for the function's values near the point \(a\).

Remember, the accuracy of a tangent line approximation increases the closer you are to the point where it's defined.

Differentiation

Differentiation is an essential tool in calculus. It's the process of finding a derivative, which represents the rate of change of a function.

• The derivative of a function provides the slope of the tangent line at any given point.
• For the cosine function, the derivative is crucial to finding how fast it changes at any point of interest.

For the cosine function \( f(x) = \cos(x) \), the derivative is \( f'(x) = -\sin(x) \). This tells us how the cosine function is decreasing or increasing at specific moments.

Interestingly, when we differentiate the cosine function, the result \(-\sin(x)\) explains its behavior in terms of slope at any point \(x\). At \(x = 0\), the slope \(-\sin(0)\) is 0, meaning that at this point, the function's rate of change is zero, indicated by a flat or horizontal tangent line.

Cosine Function

The cosine function, written as \(\cos(x)\), is one of the basic trigonometric functions. It's periodic, meaning it repeats its values in a regular cycle.

• The cosine function has a maximum value of 1 and a minimum value of -1.
• It's known to be even, meaning \(\cos(-x) = \cos(x)\), which reflects its symmetry around the vertical axis.
• Frequently used in trigonometry, physics, and engineering.

When performing linear approximation, the cosine function at \(a=0\) calculates as \(\cos(0) = 1\), giving it a local maximum at this point. Therefore, near \(x=0\), the function behaves similarly to the constant line \(y=1\), as derived from the tangent line equation.

This provides insight into how linear approximation simplifies the process of working with trigonometric functions, particularly in estimating small deviations from given points.