A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 3.1 Introduction

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.