Approximation with Differentials - Calculus I-Honors | MATH 1351, Study notes of Mathematics

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• Homework 8 due this Friday 10/26/

Approximations with Differentials

Example: approximate (^3).^198 using differentials.

Let f (x) = (^) x^1 , so f ′(x) = −x−^2.

Recall

f (x 0 + ∆x) ≈ f (x 0 ) + f ′(x 0 )dx

Let x 0 = 4 and ∆x = dx = − 0. 02

f (3.98) ≈ f (4) + f ′(4)(− 0 .02) = 14 + − 16 1 (− 0 .02) = 0 .25 + 0. 00125 = 0. 25125

Compare to calculator value:

f (3.98) ≈ 0. 2512562814

If x is the measured value and x + ∆x represents the exact value, then ∆x is the error in measurement. The difference between f (x + ∆x) and f (x) is called the propagated error at x:

∆f = f (x + ∆x) − f (x)

The relative error is ∆ff ≈ dff.

The percentage error is 100( ∆ff )%.

Newton-Raphson for Root Approximation

The Newton-Raphson method uses tangent lines to estimate roots (zeros) of equations.

Theorem: Newton-Raphson Method: To approximate f (x) = 0, start with a preliminary estimate (or guess) x 0 , and generate a sequence of estimates x 1 , x 2 , x 3 ,... using the formula:

xn+1 = xn − (^) ff ′^ ((xxn) n)^

, f ′(xn) 6 = 0

Either this sequence of approximations will approach a limit that is a root of the equation, or the sequence does not have a limit.

Extreme Values

Let f (x) be a function on an interval I containing number c.

• f (c) is an absolute maximum of f on I if f (c) ≥ f (x) for all x in I.
• f (c) is an absolute minimum of f on I if f (c) ≤ f (x) for all x in I.
• The absolute maximum and minimum are called extreme values or absolute extrema of f on I.
• Note that a function does not necessarily have a maximum or a minimum on a given interval.

Extreme Value Theorem: A function f has both an absolute maximum and an absolute minimum on any closed, bounded interval [a, b] where it is continuous.

Note: if f is discontinuous or the interval is not both closed and bounded, we cannot conclude that f has an absolute max and min.

• Suppose f is defined at c and either f ′(c) = 0 or f ′(c) does not exist. Then c is called a critical number of f , and the point P (c, f (c)) on the graph is called a critical point.
• Critical number theorem: If a continuous function f has a relative extremum at c, then c must be a critical number of f. (I.e., either the derivative is 0 or it does not exist at c.)

Procedure for finding absolute extrema

To find the absolute extrema of a continuous function f on [a, b]:

1. Compute f ′(x) and find all critical numbers of f on [a, b].
2. Evaluate f at the endpoints a and b and at each critical number c.
3. Compare the values from (2). Largest is the absolute maximum of f on [a, b]. Smallest is the absolute minimum of f on [a, b].