Problem 93 - Calculus Early Transcendentals

Question

Tangent line is \(p_{1}\) Let \(f\) be differentiable at \(x=a\) a. Find the equation of the line tangent to the curve \(y=f(x)\) at \((a, f(a))\) b. Find the Taylor polynomial \(p_{1}\) centered at \(a\) and confirm that it describes the tangent line found in part (a).

Step-by-Step Solution

Answer

Short Answer: Yes, the Taylor polynomial \(p_{1}\) and the tangent line to a function \(f(x)\) at the point \(x=a\) are the same, as they both represent the first-order linear approximation of the function around the point.

Step 1: Find the slope of the tangent line

To find the slope of the tangent line, we need to evaluate the first derivative of the function at \(x=a\). So, \(m = f'(a)\).

Step 2: Equation of the tangent line

Now that we have the slope, we can use the point-slope form of a line to find the equation of the tangent line: \(y - f(a) = f'(a)(x - a)\).

Step 3: Calculate the Taylor polynomial \(p_1\)

We're looking for the Taylor polynomial \(p_1\) centered at \(a\), so we only need the terms up to n=1 in the Taylor series formula. Therefore, \(p_1(x) = f(a) + f'(a)(x-a)\).

Step 4: Confirm that the Taylor polynomial \(p_1\) matches the tangent line in part (a)

Comparing the equation of the tangent line, \(y - f(a) = f'(a)(x - a)\), with the Taylor polynomial \(p_1(x) = f(a) + f'(a)(x-a)\), we can see that both equations are the same (by setting \(y = p_1(x)\)). This confirms that the Taylor polynomial \(p_1\) indeed describes the tangent line found in part (a).

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. The significance of the tangent line lies in its ability to provide the best linear approximation to the function at that point. Imagine you had a bumpy road, and you want to know how the road behaves exactly at a tiny point—this is where the tangent line comes into play.

Finding a tangent line involves knowing the slope at the point of tangency, which tells us how steep the curve is at that point. This slope, when combined with the coordinates of the point on the curve, helps in formulating the line equation. The tangent line plays a crucial role in calculus as it forms the basis for other methods of approximation, such as Taylor polynomials.

Differentiable

A function is said to be differentiable at a point when it possesses a derivative at that point. Intuitively, this means that the function has a tangent line at that specific point, and the graph is smooth without any sharp turns or cusps.

- Differentiability is a stronger condition than continuity.
- If a function is differentiable at a point, it is also continuous there. However, a function being continuous does not guarantee differentiability.

The importance of differentiability is that it enables us to use derivatives to understand and predict the behavior of functions locally around the point, making it possible to sketch the graph accurately or solve optimization problems.

First Derivative

The first derivative of a function tells us about the rate of change of the function as its variable changes. In simple terms, it measures the slope of the tangent line to the graph of the function at any point. For example, if you're driving a car, the speedometer shows the speed, which is effectively the first derivative of your distance with respect to time.

- A positive first derivative indicates the function is increasing at the point.
- A negative first derivative suggests the function is decreasing.
- A zero first derivative often points to a possible maximum or minimum value, known as critical points.

Calculating the first derivative involves applying differentiation rules to the function, allowing us to easily find tangent lines and analyze critical points.

Point-Slope Form

The point-slope form is a method used to write the equation of a line when we know a point on the line and the slope of the line. This is especially useful in calculus for finding the equation of a tangent line to a curve.

The formula of the point-slope form is given by:\[y - y_1 = m(x - x_1)\]Here, \(m\) is the slope, and \((x_1, y_1)\) is the point on the line. In the context of tangent lines:

• \(m\) would be the derivative of the function at the point \(x = a\).
• \((x_1, y_1)\) is the point \((a, f(a))\) on the curve.

The point-slope form provides an intuitive yet powerful way to form the equation of a line straightforwardly, especially for lines that need to model or approximate curves, as seen in the application of Taylor polynomials.