Preface
This is the note of the MIT Course 18.01.1 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.
Unit 4: Application
Definition of Critical Points
The critical points of a function $f(x)$ to be all points $x$ in the domain of $f(x)$ such that
- $f^\prime(x)=0$, or
- $f^\prime(x)$ does not exist.
The First Derivative Test
Finding Local Maxima and Minima
Suppose the function $f(x)$ is continuous at $x=a$ and has a critical point at $x=a$.
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$f$ has a local minimum at $x=a$ if $f^\prime(x)<0$ just to the left of $a$ and $f^\prime(x)>0$ just to the right of $a$.
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$f$ has a local maximum at $x=a$ if $f^\prime(x)>0$ just to the left of $a$ and $f^\prime(x)<0$ just to the right of $a$.
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The point $x=a$ is neither a local minimum nor a local maximum of $f$ if $f^\prime(x)$ has the same sign just to the left of $a$ and just to the right of $a$.
The Second Derivative Test
Suppose that $x=a$ is a critical point of $f$, with $f^\prime(a)=0$.
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If $f^{\prime\prime}(a)>0$, then $f$ has a local minimum at $x=a$.
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If $f^{\prime\prime}(a)<0$, the $f$ has a local maximum at $x=a$.
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If $f^{\prime\prime}(a)=0$, or does not exist, then the test is inconclusive, which means there might be a local minimum, or a local maximum, or neither.
Definition of Inflection Point
An inflection point is a point where the concavity of the function changes. That is the second derivative $f^{\prime\prime}(x)$ changes sign:
$f^{\prime\prime}>0$ just to the left of $x$ and $f^{\prime\prime}(x)<0$ just to the right of $x$, or vice versa.
General Strategy for Sketching Functions
- Plot
- discontinuities (especially infinite ones)
- end points (or $x\to \infty$)
- easy points ($x=0, \text{ or }y=0$)
- Plot critical points and values. (Solve $f^{\prime}(x)=0$ or undefined.)
- Decide whether $f^{\prime}<0$ or $f^{\prime}>0$ on each interval between endpoints, critical points, and discontinuities. (Valuable double check)
- Identify where $f^{\prime\prime}<0$ and $f^{\prime\prime}>0$ (concave down and concave up).
- Identify inflection points.
- Combine into graph.
Indeterminate Forms
We call $\frac 0 0$ and $\frac {\infty}{\infty}$ indeterminate forms, because when we run into them in a limit, they require further analysis to determine whether the numerator or denominator wins in the race to $0$ or $\infty$ respectively, or whether they balance out and reach some other finite limit.
L'Hôspital's Rule
If $$ \left. \begin{aligned} &f(x)\to 0 \\ &g(x)\to 0 \end{aligned} \right. \quad as\ x\to a $$ and the functions $f$ and $g$ are differentiate near the point $x=a$, then limit $$ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)} $$ provided that the left hand limit and the right hand limit exists or is $\pm\infty$. Note that
- We can replace $a$ with $a^+$ or $a^-$ and the results still hold.
- We can replace $a$ with $\pm\infty$, and the results still hold.
Other Indeterminate Forms
Other indeterminate forms $0\cdot\infty$, $\infty -\infty$, $0^0$, $1^\infty$ and $\infty^0$ should be rearranged to be of the form $0/0$ or $\infty / \infty$ in order to apply L’Hôpital’s rule.
The Extreme Value Theorem
If $f$ is continuous on a closed interval $[a,b]$, then there are points at which $f$ attains its maximum and its minimum on $[a,b]$.
Maxima and Minima
The maxima and minima will be attained at either a critical point or an end point.