Properties of differentiable functionsFermat's Theorem. Let be defined and differentiable on an open interval (a, b). If attains its absolute maximum or absolute minimum (both are called absolute extremum) at , where , then . Rolle's Theorem. If a function satisfies all the following three conditions: (1) is continuous on the closed interval , (2) is differentiable in the open interval , (3) ; then there exists at least a point such that . Lagrange's Theorem. If a function is (1) continuous on the closed interval and (2) differentiable in the open interval , then there exists at least a point such that .
Limits- indeterminate forms and L’Hospital’s Rule I. Indeterminate Form of the Type We have previously studied limits with the indeterminate form as shown in the following examples: Example 5: However, there is a general, systematic method for determining limits with the indeterminate form . Suppose that f and g are differentiable functions at x = a a nd that is an indeterminate form of the type ; that is, and . Since f and g are differentiable functions at x = a, then f and g are continuous at x = a; that is, = 0 and = 0. Furthermore, since f and g are differentiable functions at x = a, then and . Thus, if , then if and are continuous at x = a. This illustrates a special case of the technique known as
L’Hospital’s Rule.
The Bernoulli-L'Hopital Rule This rule is used for finding the ratio limits of the form Theorem 1. Suppose that there are differentiable functions f(x) and j(x) on the interval [a;b] and f(a)=j(a)=0, then having limit , there is the limit which equals . In the following examples, we will use the following three-step process:
Step 1. Check that the limit of is an indeterminate form of type . If it is not, then L’Hospital’s Rule cannot be used. Step 2. Differentiate f and g separately. [ Note: Do not differentiate using the quotient rule! ] Step 3. Find the limit of . If this limit is finite, , or , then it is equal to the limit of . If the limit is an indeterminate form of type ,then simplify algebraically and apply L’Hospital’s Rule again.
Example 6:
II. Indeterminate Form of the Type We have previously studied limits with the indeterminate form as shown in the following examples:
Example 7:
However, we could use another version of L’Hospital’s Rule.
L’Hospital’s Rule for Form Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that and . If has a finite limit, or if this limit is or
, then . Moreover, this statement is also true in the case of a limit as or as
III. Indeterminate Form of the Type
Indeterminate forms of the type can sometimes be evaluated by rewriting the product as a quotient, and then applying L’Hospital’s Rule for the indeterminate forms of type or . Example 8: IV. Indeterminate Form of the Type
A limit problem that leads to one of the expressions
, , ,
is called an indeterminate form of type . Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction. However, limits problems that lead to one the expressions , , , are not indeterminate, since the two terms work together (the first two produce a limit of and the last two produce a limit of ). Indeterminate forms of the type can sometimes be evaluated by combining the terms and manipulating the result to produce an indeterminate form of type or . Example 9:
V. Indeterminate Forms of the Types Limits of the form frequently give rise to indeterminate forms of the types . These indeterminate forms can sometimes be evaluated as follows: (1) (2) (3)
The limit on the righthand side of the equation will usually be an indeterminate limit of the type . Evaluate this limit using the technique previously described. Assume that = L. (4) Finally, . Example 10: Find .
This is an indeterminate form of the type . Let . 0. Thus, .
Taylor’s formula
Suppose we’re working with a function f(x) that is continuous and has n+1 continuous derivatives on an interval about x = 0. We can approximate f near 0 by a polynomial of degree n: • For n = 0, the best constant approximation near 0 is which matches f at 0. • For n = 1, the best linear approximation near 0 is . Note that matches f at 0 and matches at 0. • For n = 2, the best quadratic approximation near 0 is . Note that , , and match , , and , respectively, at 0. Continuing this process,
. This is the Taylor polynomial of degree n about 0 (also called the Maclaurin series of degree n). More generally, if f has n+1 continuous derivatives at x = a, the Taylor series of degree n about a is . This formula approximates f (x) near a. Taylor’s Theorem gives bounds for the error in this approximation: Taylor’s Theorem: Suppose f has n+1 continuous derivatives on an open interval containing a. Then for each x in the interval, , where the error term satisfies for some c between a and x. This form for the error , derived in 1797 by Joseph Lagrange, is called the Lagrange formula for the remainder. The infinite Taylor series converges to f, , if and only if .
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