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3.4-1 Variation

3.4-1 Variation. Many natural (physical) phenomena exhibit variation = one quantity (quantities) changing on account of another (or several) Principle is some kind of dependence What things can we think about that depend on another action/object?. Direct Variation.

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3.4-1 Variation

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  1. 3.4-1 Variation

  2. Many natural (physical) phenomena exhibit variation = one quantity (quantities) changing on account of another (or several) • Principle is some kind of dependence • What things can we think about that depend on another action/object?

  3. Direct Variation • Direct Variation = as one variable changes, the other changes at some constant rate • Y varies directly with the nthpower of x (y is proportional to the nth power of x) if: • y = kxn • K is a constant; n is a real number • D = rt is an example of direct variation

  4. The constant • In most applications, we have to determine the constant value k, given information about y and x • Example. Hooke’s Law says the force exerted by a spring on a spring scale varies directly with the distance the spring is stretched. If a 15 pound mass suspended on a string stretches the spring 6 inches, how far will a 20 pound mass stretch it?

  5. Example. Hooke’s Law says the force exerted by a spring on a spring scale varies directly with the distance the spring is stretched. If a 15 pound mass suspended on a string stretches the spring 6 inches, how far will a 20 pound mass stretch it? • y = kx

  6. Example. Write the mathematical model for the following statement. • A) S varies directly as the product of 4 and x. • B) Z varies directly with y-cubed. • C) J(x) varies directly with the nth-root of x.

  7. Inverse Variation • Inverse Variation = as one quantity increases, a second quantity decreases • y varies inversely with the nth power of x (or, y is inversely proportional to the nth power of x) if there is a constant k such that • y =

  8. Example. Supper y is inversely proportional to the 2nd power of x, and y = 9 when x = 3. What is y when x = 10? • Example. Supper y is inversely proportional to the square of x, and that y = 5 when x = 2. What is y when x = 10?

  9. Joint Variation • More than 2 variables • Z varies jointly as x and y (proportional to x and y) if there is a constant k such that • Z = kxy • Z varies jointly as the nth power of x and the mth power of y is there is a constant k such that Z = kxnym

  10. Example. Suppose z is jointly proportional to x and y, and that z = 200 when x = 10 and y = 6. What is z when x = -5 and y = 3? • Example. Suppose z is jointly proportional to the square of x and the cube of y, and that z = 500 when x = 2 and y = 27. Find z when x = 16 and y = 32.

  11. Assignment • Pg. 238 • 1-18 all

  12. Solutions

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