4.1 Related Rates

To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.
In terms of the quantities, state the information given and the rate to be found.
Find an equation relating the quantities.
Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.
Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.

4.2 Linear Approximations and Differentials

A differentiable function $y = f (x)$ can be approximated at $a$ by the linear function

$L (x) = f (a) + f' (a) (x - a) .$
For a function $y = f (x),$ if $x$ changes from $a$ to $a + d x,$ then

$d y = f' (x) d x$

is an approximation for the change in $y .$ The actual change in $y$ is

$Δ y = f (a + d x) - f (a) .$
A measurement error $d x$ can lead to an error in a calculated quantity $f (x) .$ The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by

$d y \approx f' (x) d x .$
To estimate the relative error of a particular quantity $q,$ we estimate $\frac{Δ q}{q} .$

A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.

If $f$ is continuous over $[a, b]$ and differentiable over $(a, b)$ and $f (a) = f (b),$ then there exists a point $c \in (a, b)$ such that $f^{'} (c) = 0 .$ This is Rolle’s theorem.
If $f$ is continuous over $[a, b]$ and differentiable over $(a, b),$ then there exists a point $c \in (a, b)$ such that

$f' (c) = \frac{f (b) - f (a)}{b - a} .$

This is the Mean Value Theorem.
If $f' (x) = 0$ over an interval $I,$ then $f$ is constant over $I .$
If two differentiable functions $f$ and $g$ satisfy $f^{'} (x) = g^{'} (x)$ over $I,$ then $f (x) = g (x) + C$ for some constant $C .$
If $f^{'} (x) > 0$ over an interval $I,$ then $f$ is increasing over $I .$ If $f^{'} (x) < 0$ over $I,$ then $f$ is decreasing over $I .$

If $c$ is a critical point of $f$ and $f^{'} (x) > 0$ for $x < c$ and $f^{'} (x) < 0$ for $x > c,$ then $f$ has a local maximum at $c .$
If $c$ is a critical point of $f$ and $f^{'} (x) < 0$ for $x < c$ and $f^{'} (x) > 0$ for $x > c,$ then $f$ has a local minimum at $c .$
If $f^{″} (x) > 0$ over an interval $I,$ then $f$ is concave up over $I .$
If $f^{″} (x) < 0$ over an interval $I,$ then $f$ is concave down over $I .$
If $f^{'} (c) = 0$ and $f^{″} (c) > 0,$ then $f$ has a local minimum at $c .$
If $f^{'} (c) = 0$ and $f^{″} (c) < 0,$ then $f$ has a local maximum at $c .$
If $f^{'} (c) = 0$ and $f^{″} (c) = 0,$ then evaluate $f^{'} (x)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c,$ to determine whether $f$ has a local extremum at $c .$

The limit of $f (x)$ is $L$ as $x \to \infty$ (or as $x \to − \infty)$ if the values $f (x)$ become arbitrarily close to $L$ as $x$ becomes sufficiently large.
The limit of $f (x)$ is $\infty$ as $x \to \infty$ if $f (x)$ becomes arbitrarily large as $x$ becomes sufficiently large. The limit of $f (x)$ is $− \infty$ as $x \to \infty$ if $f (x) < 0$ and $| f (x) |$ becomes arbitrarily large as $x$ becomes sufficiently large. We can define the limit of $f (x)$ as $x$ approaches $− \infty$ similarly.
For a polynomial function $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + … + a_{1} x + a_{0},$ where $a_{n} \neq 0,$ the end behavior is determined by the leading term $a_{n} x^{n} .$ If $n \neq 0,$ $p (x)$ approaches $\infty$ or $− \infty$ at each end.
For a rational function $f (x) = \frac{p (x)}{q (x)},$ the end behavior is determined by the relationship between the degree of $p$ and the degree of $q .$ If the degree of $p$ is less than the degree of $q,$ the line $y = 0$ is a horizontal asymptote for $f .$ If the degree of $p$ is equal to the degree of $q,$ then the line $y = \frac{a_{n}}{b_{n}}$ is a horizontal asymptote, where $a_{n}$ and $b_{n}$ are the leading coefficients of $p$ and $q,$ respectively. If the degree of $p$ is greater than the degree of $q,$ then $f$ approaches $\infty$ or $− \infty$ at each end.

To solve an optimization problem, begin by drawing a picture and introducing variables.
Find an equation relating the variables.
Find a function of one variable to describe the quantity that is to be minimized or maximized.
Look for critical points to locate local extrema.

L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form $\frac{0}{0}$ or $\infty / \infty$ arises.
L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form $\frac{0}{0}$ or $\infty / \infty .$
The exponential function $e^{x}$ grows faster than any power function $x^{p},$ $p > 0 .$
The logarithmic function $ln x$ grows more slowly than any power function $x^{p},$ $p > 0 .$

Newton’s method approximates roots of $f (x) = 0$ by starting with an initial approximation $x_{0},$ then uses tangent lines to the graph of $f$ to create a sequence of approximations $x_{1}, x_{2}, x_{3},… .$
Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers $x_{0}, x_{1}, x_{2},…$ does not approach a finite value or it approaches a value other than the root sought.
Any process in which a list of numbers $x_{0}, x_{1}, x_{2},…$ is generated by defining an initial number $x_{0}$ and defining the subsequent numbers by the equation $x_{n} = F (x_{n - 1})$ for some function $F$ is an iterative process. Newton’s method is an example of an iterative process, where the function $F (x) = x - [\frac{f (x)}{f^{'} (x)}]$ for a given function $f .$

If $F$ is an antiderivative of $f,$ then every antiderivative of $f$ is of the form $F (x) + C$ for some constant $C .$
Solving the initial-value problem

$\frac{d y}{d x} = f (x), y (x_{0}) = y_{0}$

requires us first to find the set of antiderivatives of $f$ and then to look for the particular antiderivative that also satisfies the initial condition.