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EXAMPLE 10 QUADRATIC FORMULA By completing the square as above we can obtain the following formula for the roots of a quadratic equation. The Quadratic Formula The roots of the quadratic equation are EXAMPLE 11 Solve the equation . SOLUTION With , , , the quadratic formula gives the solutions The quantity that appears in the quadratic formula is called the discriminant.
Cut here and keep for reference Chapter 1 ConCept CheCk answers (continued) (c) Exponential function: fsxd − 2x, fsxd − bx (d) onc ui nc: fi t Qt a drua fsxd − x2 1 x 1 1, fs xd − a 2 1b c (e) Polynomial of degree 5: fsxd − x5 1 2x4 2 3x2 1 7 (f) ononui nc: l i af Rt t a fsxd − x x 1 2, fsxd − Psxd Qsxd where Psxd nda Qsxd are polynomials 7. Sketch by hand, on the same axes, the ...
APPLIED PROJECT THE CALCULUS OF RAINBOWS 3 SOLUTIONS 1. From Snell’s Law, we have sinα= ksinβ≈4 3 sinβ ⇔ β≈arcsin 3 4 sinα .Wesubstitutethisinto D(α)=π+2α−4β= π+2α−4arcsin 3 4 sinα , and then differentiate to find the minimum: D0(α)=2−4 k 1− 3 4 sinα 2l−1/2 3 4 cosα =2− 3cosα t 1−9 16 sin2 α.Thisis0 when ...
REVIEW OF ANALYTIC GEOMETRY 3 EXAMPLE 2 Sketch the graph of the equation by first show- ing that it represents a circle and then finding its center and radius. SOLUTION We first group the -terms and -terms as follows: Then we complete the square within each grouping, adding the appropriate constants
then we know from Part 1 of the Fundamental Theorem of Calculus that Thus, has an antiderivative , but it has been proved that is not an elemen-tary function. This means that no matter how hard we try, we will never succeed in evalu-ating in terms of the functions we know. (In Chapter 8, however, we will see how to express as an infinite series.)
The analogous formula for a difference of cubes is which you can verify by expanding the right side. For a sum of cubes we have EXAMPLE 6 (a) (Equation 2; ) (b) (Equation 3; ) (c) (Equation 5; ) EXAMPLE 7 Simplify . SOLUTION Factoring numerator and denominator, we have To factor polynomials of degree 3 or more, we sometimes use the following fact.
James Stewart, Harvey Keynes, Daniel Clegg, and developer Hu Hohn Tools for Enriching Calculus (TEC) functions as both a powerful tool for instruc- tors, as well as a tutorial environment in which students can explore and review
(c) If we introduce the Bernoulli numbers, then we can write and, in general, where [The numbers are the binomial coefficients.] Use part (b) to show that, for , and therefore This gives an efficient way of computing the Bernoulli numbers and therefore the
COMPLEX NUMBERS A complex numbercan be represented by an expression of the form , where and are real numbers and is a symbol with the property that . The complex num-ber can also be represented by the ordered pair and plotted as a point in a
Formulas for the Remainder Term in Taylor Series In Section 8.7 we considered functions with derivatives of all orders and their Taylor series The th partial sum of this Taylor series is the nth-degree Taylor polynomial off at a: We can write
