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Section 3.1 Introduction

Use the following definitions, theorems, and properties to solve the problems contained in this Chapter.

Absolute Maximum and Minimum

A function \(f\) has an absolute maximum at \(c\) if \(f(c)\geq f(x)\) for all \(x\in D\text{,}\) the domain of \(f\text{.}\) The number \(f(c)\) is called the maximum value of \(f\) on \(D\text{.}\) A function \(f\) has an absolute minimum at \(c\) if \(f(c)\leq f(x)\) for all \(x\in D\text{,}\) the domain of \(f\text{.}\) The number \(f(c)\) is called the minimum value of \(f\) on \(D\text{.}\)

Local Maximum and Minimum

A function \(f\) has a local maximum at \(c\) if \(f(c)\geq f(x)\) for all \(x\) in an open interval containing \(c\text{.}\) A function \(f\) has a local minimum at \(c\) if \(f(c)\leq f(x)\) for all \(x\) in an open interval containing \(c\text{.}\)

Extreme Value Theorem

If \(f\) is continuous on a closed interval \([a,b]\text{,}\) then \(f\) attains an absolute maximum value \(f(c)\) and an absolute minimum value \(f(d)\) at some numbers \(c,d\in [a,b]\text{.}\)

Fermat's Theorem

If \(f\) has a local maximum or minimum at \(c\text{,}\) and \(f'(c)\) exists, then \(f'(c)=0\text{.}\)

Critical Number

A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that either \(f'(c)=0\) or \(f'(c)\) does not exist.

Closed Interval Method

To find the absolute maximum and minimum values of a continuous function \(f\) on a closed interval \([a,b]\text{:}\)

  1. Find the values of \(f\) at the critical numbers of \(f\) in \((a,b)\text{.}\)

  2. Find the values of \(f\) at the endpoints of the interval.

  3. The largest of the values from Step 1 and Step 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

Rolle's Theorem

Let \(f\) be a function that satisfies the following three hypotheses:

  1. \(f\) is continuous on the closed interval \([a,b]\text{.}\)

  2. \(f\) is differentiable on the open interval \((a,b)\text{.}\)

  3. \(f(a)=f(b)\text{.}\)

Then there is a number \(c \) in \((a,b) \) such that \(f'(c)=0 \text{.}\)

The Mean Value Theorem

Let \(f\) be a function that satisfies the following hypotheses:

  1. \(f\) is continuous on the closed interval \([a,b]\text{.}\)

  2. \(f\) is differentiable on the open interval \((a,b)\text{.}\)

Then there is a number \(c \) in \((a,b) \) such that \(f'(c)=\ds \frac{f(b)-f(a)}{b-a} \) or, equivalently, \(f(b)-f(a)=f'(c)(b-a) \text{.}\)

Increasing/Decreasing Test

  1. If \(f^\prime(x)>0\) on an interval, then \(f\) is increasing on that interval.

  2. If \(f^\prime(x)\lt 0\) on an interval, then \(f\) is decreasing on that interval.

The First Derivative Test

Suppose that \(c\) is a critical number of a continuous function \(f\text{.}\)

  1. If \(f'\) changes from positive to negative at \(c\text{,}\) then \(f\) has a local maximum at \(c\text{.}\)

  2. If \(f^\prime\) changes from negative to positive at \(c\text{,}\) then \(f\) has a local minimum at \(c\text{.}\)

  3. If \(f^\prime\) does not change sign at \(c\text{,}\) then \(f\) has no local minimum or maximum at \(c\text{.}\)

Concavity

If the graph of \(f\) lies above all of its tangent lines on an interval \(I\text{,}\) then it is called concave upward on \(I\text{.}\) If the graph of \(f\) lies below all of its tangents on \(I\text{,}\) it is called concave downward on \(I\text{.}\)

Concavity Test.

  1. If \(f^{\prime\prime}(x)>0\) for all \(x\in I\text{,}\) then the graph of \(f\) is concave upward on \(I\text{.}\)

  2. If \(f^{\prime\prime}(x)\lt 0\) for all \(x\in I\text{,}\) then the graph of \(f\) is concave downward on \(I\text{.}\)

Inflection Point

A point \(P\) on a curve \(y=f(x)\) is called an inflection point if \(f\) is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at \(P\text{.}\)

The Second Derivative Test

Suppose \(f^{\prime\prime}\) is continuous near \(c\text{.}\)

  1. If \(f^\prime(c)=0\) and \(f^{\prime\prime}(c)>0\) then \(f\) has a local minimum at \(c\text{.}\)

  2. If \(f^\prime(c)=0\) and \(f^{\prime\prime}(c)\lt 0\) then \(f\) has a local maximum at \(c\text{.}\)

Linear Approximation

The linear function \(L(x)=f(a)+f^\prime(a)(x-a)\) is called the linearization of \(f\) at \(a\text{.}\) For \(x\) close to \(a\) we have that \(f(x)\approx L(x)=f(a)+f'(a)(x-a)\) and this approximation is called the linear approximation of \(f\) at \(a\text{.}\)

Differential

Let \(f\) be a function differentiable at \(x\in \mathbb{R}\text{.}\) Let \(\Delta x =dx\) be a (small) given number. The differential \(dy\) is defined as \(dy=f^\prime(x)\Delta x\text{.}\)

Newton's Method

To estimate a solution, say \(x=r\text{,}\) to the equation \(f(x)=0\text{:}\)

  1. Begin with an initial guess \(x_1\text{.}\)

  2. Calculate \(\ds x_2=x_1 - \frac{f(x_1)}{f'(x_1)}\text{.}\)

  3. If \(x_n\) is known then \(\ds x_{n+1}=x_n - \frac{f(x_n)}{f^\prime(x_n)}\text{.}\)

  4. If \(x_n\) and \(x_{n+1}\) agree to \(k\) decimal places then \(x_n\) approximates the root \(r\) up to \(k\) decimal places and \(f(x_n)\approx 0\text{.}\)

Antiderivative

A function \(F\) is called an antiderivative of \(f\) on an interval \(I\) if \(F'(x)=f(x)\) for all \(x\in I\text{.}\)

Natural Growth/Decay Equation

The natural growth/decay is modelled by the initial-value problem

\begin{equation*} \frac{dy}{dt}=ky, \ \ y(0)=y_0,\ \ k\in \mathbb{R}\backslash\{0\}\text{.} \end{equation*}
Newton's Law of Cooling and Heating

is given as

\begin{equation*} \frac{dT}{dt}=k(T-T_s) \end{equation*}

where \(k\) is a constant, \(T=T(t)\) is the temperature of the object at time \(t\) and \(T_s\) is the temperature of surroundings.