AD - Applications of Differentiation Module Overview

Applications of Differentiation Module Overview

Introduction

One of the most significant applications of derivatives is finding extreme values of functions, which is fundamental when solving optimization problems. Graphing particular functions continues to be relevant as interpretations of concavity, as acceleration changing direction, and points of inflection as the speeding up or slowing down of a moving body are applied. In many practical situations, several related quantities vary together and thus the rates at which they vary are also related. Differential calculus allows us to describe and calculate these types of related rates problems.

Essential Questions

What does a related rate mean?
How do you solve related rate problems?
In what types of problems do the various differentiation rules apply?
How can derivatives be applied to solving motion problems?
Under what conditions do relative and absolute extrema occur?
How can local and global behavior of a function be predicted and explained?
How does the Extreme Value Theorem relate to graphs of continuous functions on closed intervals?
First and second derivatives of a function give what information about the function itself?
Under what conditions does Rolle's Theorem apply?
Under what conditions does the Mean Value Theorem apply?
How is the Mean Value Theorem interpreted geometrically and as a rate of change?
How does the slope of the graph relate to the graph of the derivative?
How is the increasing or decreasing behavior of f(x) related to the sign of f'(x)?
How does concavity of a graph relate to the second derivative?
How are the corresponding geometric and analytic characteristics of graphs of f(x), f'(x), and f"(x) related?
Where do points of inflection occur?
How do the notions of monotonicity and concavity assist in analyzing curves?
What is the relationship between position, velocity and acceleration?
How do you find the optimal value of a function with given conditions?
How does the zooming capability of graphing technology assist in revealing local linearity?
How are linear approximations and differentials used to approximate curves and solve problems?

Key Terms

The following key terms will help you understand the content in this module.

Concavity - The situation describing whether the curve y=f(x) bends upward (concave upward) or downward (concave downward). If f(x) is a differentiable function on an open interval (a, b) and L is a tangent line to y = f(x), then the graph of f is concave upward on (a, b) when it lies above all of its tangents or concave downward on (a, b) when it lies below all of its tangents.

Critical number - A number c in the domain of a differentiable function f(x) such that either f'(c) = 0 or f'(c) does not exist.

Differential - An increment of a variable that is any nonzero real number.

Extrema - (plural form of extremum) The minimum and maximum of a function on an interval, which may be either local or global.

First Derivative Test - Suppose that c is a critical number of a continuous function f. (i) If f' changes from positive to negative at c, then f has a local maximum at c. (ii) If f' changes from negative to positive at c, then f has a local minimum at c. (iii) If f' does not change sign at c, then f has no local maximum or minimum at c.

Global (Absolute) maximum/minimum value - If f is a function with domain D, then f(c) is the global (absolute) maximum value of f on D provided f(c) ≥ f(x) for all x in the domain D and f(c) is the global (absolute) minimum value of f on D provided f(c) ≤ f(x) for all x in the domain D.

Horizontal asymptote - The line y=L is a horizontal asymptote of the curve y=f(x) if either $LaTeX: \lim_{x \to \infty }=L$ $\lim_{x \to \infty }=L$ or $LaTeX: \lim_{x \to -\infty }=L$ $\lim_{x \to -\infty }=L$

Linear approximation - The equation of the tangent line to a curve (a, f(a)) and given by f(x) ≈ f(a) + f\'(a)(x - a).

Local linearity - The graph of a function appears to coincide with its tangent line near x = a and has a well-defined slope at x = a.

Mean Value Theorem - Suppose that the function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Then, for some c in (a,b), $LaTeX: f'\left(c\right)=\frac{f\left(b\right)=f\left(a\right)}{b-a}$ $f'\left(c\right)=\frac{f\left(b\right)=f\left(a\right)}{b-a}$ .

Monotonicity - Characteristic of a function related to its consistently increasing or decreasing behavior.

Optimization - The determination of maximal or minimal values of a function, often subject to constraints.

Point of inflection - A point P on a continuous curve y=f(x) at which the graph changes direction of concavity, i.e. from concave upward to concave downward or vice versa.

Related rates - Rates of change of two or more related variables that are changing with respect to time.

Relative (Local) maximum/minimum - A function f has a relative (local) maximum at c if f(c) ≥ f(x) where x is near c and a relative (local) minimum at c if f(c) ≤ f(x) where x is near c.

Rolle's Theorem - Suppose that the function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and that f(a) = f(b). Then, for some c in (a,b), f'(c) = 0.

Second Derivative Test - Suppose that the function f is differentiable on an open interval containing the critical point c and f'' is continuous near c. (i) If f'(c)=0 and f''(c) > 0, then f has a local minimum at c. (ii) If f'(c)=0 and f''(c) < 0, then f has a local maximum at c.

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