First and Second Derivatives with Taylor Expansions

1. First and Second Derivatives

The first derivative of a function f(x), denoted f'(x), measures the rate of change of f(x) with respect to x. It tells us whether the function is increasing or decreasing at a given point.

The second derivative, f''(x), measures the rate of change of the first derivative. It tells us about the concavity of the function (whether it's curving upward or downward) and helps identify points of inflection.

 Example:

Consider f(x) = x^2.

- First derivative: f'(x) = 2x. This tells us that the slope of f(x) at any point x is 2x.

- Second derivative: f''(x) = 2. Since f''(x) > 0, the function is concave upward everywhere.

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2. Taylor Expansions

A Taylor expansion approximates a function f(x) near a point x = a using its derivatives. The idea is to express f(x) as a polynomial, which is easier to work with.

The Taylor seriesof f(x) around x = a is:

f(x) = f(a) + f'(a) (x - a) + (f''(a) / 2!) (x - a)^2 + (f'''(a) / 3!) (x - a)^3 + ....

For many practical purposes, we only need the first few terms:

f(x) =approx f(a) + f'(a)(x - a) + (f''(a) / 2) (x - a)^2

This is called the second-order Taylor approximation.

 Example:

Let’s approximate f(x) = sin(x) near x = 0.

- f(0) = sin(0) = 0

- f'(x) = cos(x), so f'(0) = 1

- f''(x) = -sin(x), so f''(0) = 0 

The second-order Taylor expansion is:

sin(x) =approx 0 + 1 (x - 0) + (0 / 2) (x - 0)^2 = x

So, near x = 0, sin(x) =approx x.

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3. Making Things Rigorous

To make these ideas rigorous, we need to:

1. Define derivatives precisely: Use limits to define f'(x) and f''(x).

   f'(x) = lim h -> 0 (f(x + h) - f(x)) / h

   f''(x) = lim h -> 0 (f'(x + h) - f'(x) / h

2. Prove Taylor's theorem: Show that the Taylor series approximates f(x) well near x = a, with an error term (remainder) that goes to zero as you include more terms.

   The remainder term R n(x) in the Taylor series is:

   R n(x) = (f^(n+1) (c) / (n+1)!) (x - a)^(n+1)

   where c is some point between x and a. This ensures the approximation is accurate.

3. Apply these tools: Use derivatives and Taylor expansions to analyze functions, optimize them, or solve problems.

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4. Example: Optimization

Suppose you want to find the minimum of f(x) = x^3 - 3x^2 + 4

1. Compute the first derivative:

   f'(x) = 3x^2 - 6x

2. Find critical points by solving f'(x) = 0:

   3x^2 - 6x = 0 implies x = 0 or x = 2

3. Use the second derivative to determine if these points are minima or maxima:

   f''(x) = 6x - 6

    - At x = 0: f''(0) = -6 < 0, so x = 0 is a local maximum.

   - At x = 2: f''(2) = 6 > 0, so x = 2 is a local minimum.

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5. Key Takeaways

- Derivatives measure rates of change and concavity.

- Taylor expansions approximate functions using polynomials.

- Rigor comes from precise definitions (limits) and error analysis (remainder terms).