IMSA MI-4 Prob. Set #6 DUE: Friday, October 5th Fall 17

1) Solve the equation ( give exact solutions):

a. z6  7z3  8  0 b. graph the solutions to (a) in the complex plane

2) Find the domain for each of the functions:  3  4 x  1 a. f ( x )  l o g x   b. f ( x )   x  2  c o s x

3) Solve for x:

a. l o g 2 ( l o g 3 ( l o g 4 x ) )  0 b. x l o g 1 0 x  1 0 0 0 0

4) Find the exact value of:

a. (log4 6)(log 5 7)log 6 (8)...... (log 126 128)(log 127 129) 1 1 b.  l o g 4 7 8 4 l o g 7 7 8 4

5) How many even 5-digit integers can be formed using only the digits 2, 3, 6, 7, 8 satisfying the following conditions, if no repetition is allowed? (Each of these is a separate problem) a. There are no additional restrictions on the digits. b. The middle of the five digits must be odd. c. The integers must be greater than 70,000. d. The integers must be a multiple of 3 AND not all the digits need be used to form the number.

6) Exploration: Use Desmos, Mathematica, or another computer graphing program do the following.

sin x a) g(x)  2 is periodic but not sinusoidal. State the period, domain, and range of this function. Attach a printout of your graph. sin(ax) b) h(x)  2 . Examine how the graph changes as a changes. This would be a good time to use a parameter a and animate it with a slider. Generalize the effect changing a has on the graph. sin x c) j(x)  b . Generalize the effect of changing b. Include any restrictions on b. sin(ax) d) f (x)  b . State the period, domain, and range of f. Assume b > 0 .

7) Tomatoes are on sale, so you buy 60 pounds of them at the local market for your restaurant. Each tomato is actually 98% water. Several days later the tomatoes have dried out to the point that each is now 95% water. How much do all of your tomatoes weigh at this time?

page 1 F08 8) Let a, b, c be distinct real numbers, in geometrical progression, whose sum is 84. Determine a, b, and c if they are the 1st, 3rd, and 13th term of an arithmetical sequence.

3 9) Verify that for an equilateral triangle with side s the radius of the inscribed circle is s and the 6 3 radius of the circumscribed circle is s . 3

10) Find the sum of all the positive integers less than 1000 that are not divisible by 17.

11) Find the cubic polynomial P(x) such that 1 P(1) + P(2) + . . . + P(n) = (n + 1) n (n –1) (n – 2) 4

for all positive integers n. Completely factor P (x).

n 2 12) Write Sn =(2 k - 5 k ) in terms of n and write the result as a single fraction. k =1

13) A 15" (radius) wheel on a car is rotating at 900 rpm (revolutions per minute). Find the speed of the car in miles per hour. Give your answer to one decimal place.

14) NC (Do not use key.) Evaluate the sums:

2017 2017 n n n 骣i a) (-1) n b) i c) 琪 n=1 n=1 n=1 桫2

15) Find the inverse of each function and give its domain: 3 - x a. f ( x )  5 e 1  x b. g( x ) = c. h( x )= - 2 + cos(3 x - 1) x - 5 (Assume x is in an appropriate domain for inverse function)

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