More Derivatives

Second Derivatives 2D 3D Parametric Equations Contents

The second derivative of a function is the derivative of the derivative.

For a function f(x), the second derivative is written either as f''(x) or as d²f/dx².

[D]

Exercises

1. Find the second derivative for each of the following functions, and check using the demo:

f(x) = 0
f(x) = 1
f(x) = x
f(x) = x + 5
f(x) = x²
f(x) = x² + 2x + 3
f(x) = sin(x)
f(x) = cos(2x)

2. Try various polynomial functions for f(x). How do the degrees of f'(x) and f''(x) relate to the degree of f(x)?

Convexity and Concavity 2D Top of Page Contents
(Corresponds to Hessian in 2D)

The second derivative of a function is the derivative of the first derivative.

If f'(x) itself is differentiable, then its derivative with respect to x is denoted f''(x). If f''(x) > 0, then f'(x) is increasing and the shape of the graph is concave upward (or convex downward). If f''(x) < 0, then d'(x) is decreasing and the graph is concave downward (or convex upward). A point where the sign of the second derivative changes sign is called an inflection point.

[D]

Exercises

1. Using the demo, investigate the concavity of the following functions. For which intervals is the graph concave upward? For which intervals is it concave downward? For which intervals is there no concavity?:

f(x) = x + 0.3
f(x) = x²
f(x) = x² + x + 1
f(x) = -3x²
f(x) = x³ + 3x² + 3x + 1
f(x) = cos(x)

2. Describe the concavity of any function of the form f (x) = ax + b.

3. Now consider a function of the form f(x) = ax² + bx + c. How do the values of a, b, and c affect the concavity of f(x)?

Derivative Test

The sign of the second derivative can help determine whether a critical point is a maximum or a minimum.

If f'(x₀) = 0 and f''(x₀) > 0, then the function graph is convex upward at that point and the point must be a relative minimum.
Similarly if f'(x₀) = 0 and f''(x₀) < 0, then the function graph is convex downward at that point and the point must be a relative maximum. If f'(x₀) = 0 and f''(x₀) = 0, then this test is inconclusive. If f''(x₀) > 0 on one side of x₀ in a neighborhood of x₀ and f''(x₀) < 0 on the other side, then x₀ is called a horizontal inflection point.

[D]

Exercises

1. Use the second derivative test to determine whether each of the following critical points is a maximum or a minimum, and check using the demo:

f(x) = x², x = 0
f(x) = x² - x⁴, x = 0
f(x) = x³ - 3x, x = 1
f(x) = e^x - x, x = 0
f(x) = 1/x² + x, x = 2^1/4

2. Describe what the graphs of the function, derivative, and second derivative look like at local maxima. What about local minima?

Taylor Series 2D 3D Top of Page Contents

Taylor series are polynomials that approximate functions.

The existence of Taylor series is based on the fact that one can use the function value and first derivative, second derivative, third derivative, etc. at some x value x₀ to predict what the values of a function f(x) should be for different x values some distance from the point x₀.

If we know up to the nth derivative of f(x) as well as the function value at x₀, then we can construct a Taylor polynomial of degree n.

The first-order Taylor approximation P₁(x) is simply the tangent line at x₀, P₁(x) = f(x₀)+(x-x₀)f '(x₀).

The second-order Taylor approximation P₂(x) is the parabola that has the same function value and first and second derivatives as f(x) at the point x₀. The equation for this approximation is as follows:

T₂(x) = f(x₀)+f'(x₀)(x-x₀)+(1/2)f''(x₀)(x-x₀)².

The nth order Taylor approximation is

Σ_{i = 0}ⁿ[f⁽ⁱ⁾(x₀)/i!]*(x - x₀)ⁱ

Where f⁽ⁱ⁾(x) indicates the ith derivative of f(x).

[D]

Exercises

1. For a polynomial function of degree n, what condition is needed for a Taylor approximation to be exactly the same as the function itself?

2. Describe the behavior of the second order Taylor approximation for the function f(x) = |x² - 1|.

Constrained Maxima & Minima 2D 3D Top of Page Contents

Suppose we want to find the maximum and minimum values of some function f(x) that satisfy the condition that some function g(x) is equal to c.

If g(x) is only equal to c for a finite number of x values (which is true except for a few special cases, such as g(x) being a constant function equal to c), then the solution is simply to see which of these values of x yields the maximum and minimum values of f(x).

[D]

Exercises

1. Find the constrained maxima and minima in each of the following situations:

f(x) = x, g(x) = 2x, c = 1
f(x) = x, g(x) = x³ - x, c = 0
f(x) = sin(x), g(x) = x², c = π²/16

2. The demo begins with the example g(x) = √(x² + f(x)²). What is the geometrical significance of g(x)?