3. Consider the statement R=(Ex € Z) [(r² = (1 + 1)2) = (23 EZ)]. (a) Write the negation of R without using the symbol . You may use the symbols # and/or &, if necessary. NO explanation is needed. (b) Which statement is true, Ror-R ? Circle the correct answer. NO explanation is needed. R is TRUE -R is TRUE
Homework Help for Mathematics (page 866)
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1. (16 pts) Let f be the function with domain [0, 6] given by the graph below. y=f(x) 3 4 5 6 Use the graph off to answer the following questions about f. (a) List the critical point(s) of f on the interval (0,6). Write "none" if appropriate. (b) List the inflection point(s) of f on the interval (0,6). Write "none" if appropriate. Use the graph off to answer the following questions about the derivative f' of f. (c) List the intervals(s) on which f' is negative. Write "none" if appropriate. (d) List the intervals(s) on which f' is decreasing. Write "none" if appropriate. (e) List the x-coordinates(s) at which f' has a local minimum. Write "none" if appropriate.
75 9. A marketing researcher has determined that when a product sells at a dollars per unit, the number of units sold per week is given by the function q = - + 80(5m - 2) where m>O is the total manufacturing cost in dollars per unit and m < x < 5m. 2 m (a) Find the profit function P(x) (in dollars per week) and simplify it. (5 points) (b) Use calculus to find the selling price x that maximizes weekly profit. Sufficiently justify your solution. [6 points] (c) Use your answer to part (b) to determine the manufacturing cost m that yields a maxi- mum weekly profit of 3, 455 dollars. [4 points)
2 11. Let f(c) be a continuous function defined on (a, b) whose range is (a, b). Show that there is a point on the graph of f that intersects the line y = r. That is, show that there exists a number cela, b] such that f(c) = c. Solution: If f(a) = a, then we can take c= a. If f(0) = b, then we can take c=b. Assume that f(a) +a and f(b) + b and consider g(1) = f(I) - I. Observe that since the range of f is [a, b] and f(a) +a, we have that f(a) > a. Hence, g(a) = f(a) - a>0. Similarly, observe that since the range of f is [a, b] and f(b) + b, we have that f(b)
16. A bag of candy contains 3 chocolate bars, 2 peppermint candies and 2 lollipops. A child is allowed to select three candies one after the other and without replacement except that if a chocolate bar is chosen no further selection is made. How many different ordered selections are possible? a) 12 b) 13 c) 18 d) 10 e) 14 0 9 B) none of the others
3. (15 pts) (I) The graph of the functions (t) =- - t +1 is given below. s=s(t) (a) In the figure above draw a secant line joining the points (o, s (0)) and (2, s (2)). (b) Find the slope, msec, of this secant line. (c) Draw a tangent line to the curve s = s(t) at the point where t = 2. (d) Is the slope of this tangent line, mtan, positive or negative ? 12 - is the position of an object moving along a line (II) Suppose s (t) = — at time to. (a) How is the average velocity, Vav, between the times t = 0 and t = 2 related to the slope msec from the part (I b)? (b) Is the instantaneous velocity, Vinst, at t = 2 positive or negative? Explain.
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