91Ó°ÊÓ

Linear approximation is a method used in calculus to estimate the value of a function around a specific point. It utilizes the concept of a tangent line to approximate the function. For a function \(f(t)\) near a point \(t = a\), the linear approximation is expressed as:

• \(p_1(t) = f(a) + f'(a)(t-a)\)

This equation uses the function's value and its slope at \(a\), detailed by the derivative, to create an approximate equation of the tangent line. Linear approximation is useful for simplifying calculations but is most accurate near \(t = a\). It becomes less precise as you move away from this point. It essentially "flattens" the curve to a straight line, making it easier to work with for small intervals.

Derivatives play a crucial role in calculus, especially when it comes to finding linear approximations. The derivative of a function measures how the function's value changes as its input changes. It's essentially the slope of the tangent line at any given point. For the function \(R(t) = \frac{1}{t}\), the derivative is determined by using the power rule, resulting in:

• \(R'(t) = -\frac{1}{t^2}\)

This expression gives the rate of change of the function at any point \(t\). Calculating the derivative at a specific value, such as \(t = 0.5\), helps in forming the linear approximation. In essence, the derivative allows us to "capture" the function's behavior precisely at a given point, making it an essential tool for approximation and other applications in calculus.

The tangent line is a straight line that just "touches" the function at a given point. It doesn't cross it at that location, and it's essentially the line that best represents the slope of the function at that precise point. For the function \(R(t) = \frac{1}{t}\), the tangent line at \(t = 0.5\) is formulated using the derivative and the value of \(R(t)\) at this point. The formula for the tangent line, also the linear approximation, is:

• \(p_1(t) = R(a) + R'(a)(t-a)\)

Substituting the known values gives us the linear approximation: \(p_1(t) = 4 - 4t\). This line provides a simple, yet effective approximation of \(R(t)\) near \(t = 0.5\), and is instrumental in estimating the function's values with ease.

Function evaluation is the process of determining the value of a function at a specific input. In the exercise, we need to evaluate \(R(t) = \frac{1}{t}\) and its linear approximation \(p_1(t)\) at \(t = 0.7\). This involves substituting \(t = 0.7\) into the functions:

• \(R(0.7) = \frac{1}{0.7} \approx 1.4286\)
• \(p_1(0.7) = 4 - 4 \times 0.7 = 1.2\)

Function evaluation helps us compare the actual function value with the estimated values from the linear approximation. This comparison illustrates how close or far apart these values are and demonstrates the approximation's accuracy at \(t = 0.7\). It also highlights the linear approximation's limitation in accuracy as you move away from the tangent point.