More Applications of Differentiation

Suppose you have an equation relating to time-dependent quantities. If you differentiate it with respect to time, you end up with an equation relating the derivatives of the quantities.

Method of related rates
Suppose you have two time-dependent quantities, \(A = A(t) \) and \( B = B(t) \), which are linked by an equation \(F(A(t),B(t)) = 0\). If \(A'(t) \) is given and you want to find \(B'(t) \), differentiate implicitly both sides of the equation, \( \frac{d}{dt} F(A(t),B(t)) = 0 \). You will then get an expression involving \(A'(t)\) and \(B'(t)\), allowing you to solve for one of them in terms of the other.

In applications one often has to solve equations which can not be solved analytically. One then usually turns to numerical solution techniques, of which Newton's method is the most famous.

Newton's method
Suppose you want to find a root of the equation \(f(x) = 0\). Pick some number \(x_0\), and consider the sequence \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.\] If \(x_n\) approaches some number \(r\) when \(n \to \infty\), then \(f(r) = 0\) (at least if \(f/f'\) is continuous at the point \(r\)).

If \(\lim f(x) = L\) and \(\lim g(x) = M\), then a limit rule tells us that, as long as \(M \neq 0\), we have that \(\lim f(x)/g(x) = L/M\). Of course, if \(M=0\) and \(L \neq 0\), then the limit is one of \(\pm \infty\). But what if both \(L=M=0\) ? Then the limit is a "0/0" expression, which is referred to as an indeterminate form. Although these limits can sometimes be evaluated by e.g. factoring the numerator and denominator, we now present a general way to attack such limits.

Theorem 3: The first l'Hôpital Rule
Suppose that \(f\) and \(g\) are differentiable on the interval \((a,b)\). Suppose also that \[\begin{align} & (i) \quad \lim_{x \to a+} f(x) = \lim_{x \to a+} g(x) = 0, \\ & (ii)\quad \lim_{x \to a+} \frac{f'(x) }{g'(x)} = L. \end{align}\] Then also \[\lim_{x \to a+} \frac{f(x)}{g(x)} = L. \] Similar results holds if the limit is a usual two-sided limit, a one-sided limit from the left, or if it is a limit at infinity.

The second rule helps dealing with the indeterminate form "\(\infty/\infty\)".

Theorem 4: The second l'Hôpital Rule
Suppose that \(f\) and \(g\) are differentiable on the interval \((a,b)\). Suppose also that \[\begin{align} & (i) \quad \lim_{x \to a+} g(x) \pm \infty, \\ & (ii)\quad \lim_{x \to a+} \frac{f'(x) }{g'(x)} = L. \end{align}\] Then also \[\lim_{x \to a+} \frac{f(x)}{g(x)} = L. \] Similar results holds if the limit is a usual two-sided limit, a one-sided limit from the left, or if it is a limit at infinity.

When trying to evaluate a limit one often comes across other indeterminate forms like \(0^0\) or \(0 \cdot \infty\). The strategy then is to rewrite the limit as one of the indeterminate form \(0/0\) or \(\infty/\infty\), one use the corresponding l'Hôpital Rule.

First we define what we mean by a global and a local extreme value. We state the definitions for maxima; the definitions for minima are analogous.

Definition 1: Absolute maximum value
A function \(f\) has an absolute maximum value \(f(x_0)\) at the point \(x_0\) in its domain if \(f(x) \le f(x_0)\) for every \(x\) in the domain of \(f\).

Definition 2: Local maximum value
A function \(f\) has a local maximum value \(f(x_0)\) at the point \(x_0\) in its domain if there exists a number \(h>0\) such that \(f(x) \le f(x_0)\) for every \(x\) in the domain of \(f\) such that \(|x-x_0|<h\).

Recall that \(x_0\) is a critical point of \(f\) if \(f'(x_0) = 0\). We further define a point \(x_0\) to be a singular point if \(f'(x_0)\) is not defined. Using this terminology, we can give a theorem which characterizes the points at which a (local) extreme value can occur.

Theorem 6: Locating extreme values
If the function \(f\) is defined on an interval \(I\) and has a local maximum or minimum at the point \(x_0\), then \(x_0\) is either a critical point, a singular point, or an endpoint of \(I\).

While the previous theorem tells you at what points extreme values can occur, it doesn't say if the point in question is a local maximum or a local minimum, or neither. But once these points are found the following theorem can help with this. We give the result for maxima; the result for minima is analogous.

Theorem 7: The First Derivative Test for Maxima
PART I. Testing interior crictical points and singular points.
Suppose that \(f\) is continuous at \(x_0\), and \(x_0\) is not an endpoint of the domain of \(f\).
(i) If there exists an open interval \((a,b)\) containing \(x_0\) such that \(f'(x)>0\) on \((a,x_0)\) and \(f'(x)<0\) on \((x_0,b)\), then \(f\) has a local maximum at \(x_0\).

PART II. Testing endpoints.
Suppose \(a\) is a left endpoint of the domain, and \(f\) is right continuous at \(a\).
(ii) If \(f'(x)<0\) for some interval \((a,b)\), then \(a\) is a local maximum.

Suppose \(b\) is a right endpoint of the domain, and \(f\) is left continuous at \(a\).
(iii) If \(f'(x)>0\) for some interval \((a,b)\), then \(b\) is a local maximum.

We have seen that given a function \(f\) differentiable at the point \(a\), the linearization \(L(x) = f(a)+f'(a)(x-a)\) provides a good linear approximation of \(f\) near \(a\). be defined as the unique polynomial of degree 1 such that \(L(a) = f(a)\) and \(L'(a) = f'(a)\). Phrased in this way, it is natural to ask how good an approximation we can achieve if we allow for polynomials of higher degrees matching the values of the higher order derivatives of \(f\) at the point \(a\). These approximating polynomials are called Taylor polynomials.

Taylor polynomials
Suppose \(f\) is \(n\) times differentiable at \(a\). The \(n\)'th order Taylor polynomial for \(f\) about \(a\) is the polynomial \[P_n(x) = f(a) + \frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n.\] \(P_n\) is the unique polynomial of degree at most \(n\) such that \(f(a) = P_n(a), \ f'(a) = P_n'(a), \ldots ,f^{(n)}(a) = P^{(n)}(a).\) \(P_1\) is just the linearization of \(f\) about \(a\).

The following theorem gives precise estimates for how good an approximation the Taylor polynomials about \(a\) are to the function near \(a\).

Theorem 12: Taylor's Theorem
If the \((n+1)\)st derivative \(f^{(n+1)}\) is defined for all \(t\) in an interval containing \(a\) and \(x\), then the error \(E_n(x) = f(x)-P_n(x)\) satisfies \[E_n(x) = \frac{f^{(n+1)}(s)}{(n+1)!}(x-a)^{n+1}\] for some number \(s\) between \(a\) and \(x\).