Final Assignment prior to Test #1

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• Laws of Exponents

• Solving Equations of the form x ^ a = b or x ^ (a/b) = c

• Exercises 1-4

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Laws of Exponents

You should be able to use these laws of exponents:

• (x ^ a) ^ b = x ^ (ab)

• e.g., (x ^ 2) ^ 3 = x^2 * x^2 * x^2 = x*x * x*x * x*x = x^6

• (x ^ a) (x ^ b) = x ^ (a + b)

• e.g., (x^2)(x^3) = x*x * x*x*x = x^5

• x^0 = 1

• (x^a) * (x^0) = x^(a+0) = x^a. Since (x^a) * (x^0) = x^a, x^0 gotta be 1.

• x ^ -a = 1 / (x^a)

• since (x^-a) (x^a) = x^(-a + a) = x^0 = 1, (x^-a) (x^a) = 1 and x^-a = 1 / x^a

Solving Equations of the form x ^ a = b or x ^ (a/b) = c

You will be expected to be able to solve equations of these forms.

• to solve x^a = b, take the 1/a power of both sides to get

• (x^a) ^ (1/a) = x^(a * 1/a) = x^1 = x.

• (x^a) ^ (1/a) = b^(1/a), or x = b^(1/a).

• To solve x ^ (a/b) = c, take the reciprocal power of both sides to get

  ( x ^ (a/b) ) ^ (b/a) = c ^ (b/a), or x = c ^ (b/a).

Exercises 1-4

These exercises are related to the above laws of exponents, and to the topics on quadratic functions highlighted on the

1.  Solve the equations

• x ^ 3 / 17 = 58
• (3 x) ^ -2 = 19
• 4 x ^ -.5 = 7
• 14 x ^ (2/3) = 39
• 5 ( 3 x / 8) ^ (-3/2) = 9

2.  If a(n+1) = a(n) + .5 n, with a(0) = 2, then

• Find a(1), a(2), a(3), a(4) and a(5).
• Graph a(n) vs. n. Use DERIVE (easiest) or another method (e.g., simultaneous equations, etc.) to fit a quadratic to three of your four graph points, and
• show that this function fits all five points exactly.

3.  If f(x) = .3 x^2 - 4x + 7, then evaluate f at x = 0, .4, .8, 1.2, 1.6 and 2.0.

• Show that the second difference of your sequence is constant.
• Find the average rate of change from each x value to the next.

• Graph your average rates vs. the midpoint x values (e.g., the midpoint x value between 0 and .4 is .2; midpoint between x = 1.6 and x = 2.0 is x = 1.8; for each interval graph the slope vs. the midpoint x value).
• Fit a linear function to your graph and show that it accurately predicts every point.

• Write your average rates as a sequence and show that the first difference of your sequence is constant.
• Explain in terms of your calculations the meaning of the statement "the rate of change of a quadratic sequence changes at a constant rate".

4.  If f(x) = a x^2 + b x + c, then what symbolic expression stands for the average slope between x = h and x = k?

• Simplify this expression and show that it is a linear function of x (try to simplify it algebraically, then use DERIVE to simplify it).