Tangent line is p₁ 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. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

Verified step by step guidance

1. Recall that the equation of the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\) can be found using the point-slope form of a line. The slope of this tangent line is given by the derivative of \(f\) at \(x = a\), which is \(f'(a)\).
2. Write the equation of the tangent line using the point-slope form: \[y - f(a) = f'(a)(x - a)\] This expresses the line passing through \((a, f(a))\) with slope \(f'(a)\).
3. Recall the first-degree Taylor polynomial (linear approximation) of \(f\) centered at \(a\) is given by: \[p_1(x) = f(a) + f'(a)(x - a)\] This polynomial approximates \(f(x)\) near \(x = a\).
4. Compare the equation of the tangent line from part (a) with the Taylor polynomial \(p_1(x)\). Notice that both have the same form, representing a line with slope \(f'(a)\) passing through \((a, f(a))\).
5. Conclude that the Taylor polynomial \(p_1(x)\) centered at \(a\) indeed describes the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\), verifying the relationship between the two.

Key Concepts

Derivative and Differentiability

The derivative of a function at a point measures the instantaneous rate of change or slope of the function's graph at that point. Differentiability at x = a means the derivative f'(a) exists, allowing us to find the slope of the tangent line to the curve y = f(x) at (a, f(a)).

Equation of the Tangent Line

The tangent line to the curve y = f(x) at x = a is a straight line that touches the curve at (a, f(a)) with slope f'(a). Its equation is given by y = f(a) + f'(a)(x - a), representing the best linear approximation of the function near x = a.

Taylor Polynomial of Degree One (Linear Approximation)

The first-degree Taylor polynomial p₁ centered at a approximates the function near x = a using p₁(x) = f(a) + f'(a)(x - a). This polynomial matches the value and slope of f at a, thus describing the tangent line and providing a linear approximation of the function.