Problem 81 - Calculus Single Variable

Question

Use a central difference quotient to approximate \(f^{\prime}(c)\) for the given \(f\) and \(c .\) Plot the function and the tangent line at \((c, f(c))\). $$ f(x)=x \operatorname{arcsec}\left(1+x^{4}\right), \quad c=2.1 $$

Step-by-Step Solution

Answer

The approximate derivative at \( c = 2.1 \) is found using the central difference quotient with \( h = 0.01 \). The tangent line is plotted using this derivative.

Step 1: Understand the Central Difference Quotient

The central difference quotient approximates the derivative of a function at a point \( c \) using two nearby points. The formula is given by: \[ f'(c) \approx \frac{f(c+h) - f(c-h)}{2h} \] where \( h \) is a small number.

Step 2: Choose a Value for h

Select a small value for \( h \). A common choice is \( h = 0.01 \). This ensures a good balance between accuracy and avoiding round-off errors.

Step 3: Calculate f(c + h) and f(c - h)

Substitute the value of \( h \) into the function: - \( f(c + h) = (2.1 + 0.01) \cdot \operatorname{arcsec}(1 + (2.1 + 0.01)^4) \)- \( f(c - h) = (2.1 - 0.01) \cdot \operatorname{arcsec}(1 + (2.1 - 0.01)^4) \) Calculate these expressions using a calculator or computational software.

Step 4: Apply the Central Difference Quotient

Use the calculated values to approximate the derivative: \[ f'(c) \approx \frac{f(2.11) - f(2.09)}{2 \times 0.01} \] Compute this expression to find an approximate value for \( f'(2.1) \).

Step 5: Plot the Function and the Tangent Line

Use the approximate value of \( f'(2.1) \) to find the equation of the tangent line at \( c = 2.1 \). The equation of the tangent line is given by:\[ y = f(c) + f'(c)(x - c) \] Plot the function \( f(x) = x \operatorname{arcsec}(1 + x^4) \) and add the tangent line calculated at \((2.1, f(2.1))\).

Key Concepts

Derivative Approximation

Approximating the derivative of a function at a specific point can be quite fundamental in calculus, especially when you can't easily derive it analytically. The central difference quotient is a method used to approximate the derivative of a function at a point. This method is particularly useful because it uses values on either side of the target point, providing a symmetrical calculation. The formula for this approximation is given by:

• \( f'(c) \approx \frac{f(c+h) - f(c-h)}{2h} \)

Here, \( h \) is a very small number, often set around 0.01, to ensure a good balance between approximation accuracy and numerical stability. By calculating \( f(c+h) \) and \( f(c-h) \) using this small \( h \), you can approximate the derivative without diving into complex calculus.

Tangent Line

In calculus, the tangent line to a function at a given point is an essential concept that helps illustrate how the function behaves locally around that point. The tangent line can be interpreted as the line that just 'touches' the function at a given point, mimicking its slope there. To find a tangent line for the function \( f(x) \) at a point \( c \), you need:

• The function's value at that point, \( f(c) \).
• The derivative value, \( f'(c) \), which represents the slope of the tangent line.

Once these elements are known, the equation of the tangent line in point-slope form is given by:

• \( y = f(c) + f'(c)(x - c) \)

This equation captures the linear behavior of the function at that specific point and gives insight into how quickly the function is changing.

Arcsec Function

The arcsecant, or \( \operatorname{arcsec} \), is the inverse of the secant function. While it's not as commonly encountered as functions like sine or cosine, understanding it is crucial in some areas of calculus and trigonometry.The secant function itself is related to cosine and defined as \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, the arcsec function is concerned with inherently reciprocal relationships. To visualize, if \( y = \sec(x) \), then \( x = \operatorname{arcsec}(y) \). It maps a value back to an angle whose secant is that value. Arcsec is particularly noteworthy in applications involving angles beyond the conventional range of sine and cosine, expanding our problem-solving toolkit.

Plotting Functions

Plotting a function provides a visual representation of its behavior, crucial for a well-rounded understanding. Examining both the function and its tangent line visually offers insights that are tough to capture numerically alone.When plotting the function \( f(x) = x \operatorname{arcsec}(1 + x^4) \), make sure to consider the range of interest. Here, the point \( c = 2.1 \) is of particular interest.First, graph the function over a suitable range to capture general behavior. Then, superimpose the tangent line found at \( c \) to illustrate how closely it aligns with the curve.Using software tools or graphing calculators can simplify this process. They allow for a more accurate and dynamic view, enhancing learning and problem comprehension.