Algebra 1. Convert each fraction to a decimal, indicating clearly the repeating part, if any. (These exercises involve numbers too large to be done exactly on most calculators; nevertheless they can be readily done by hand, or with extra thought.) 1836942754321 54 11 (a) (b) (c) 25 37 17 2. Evaluate 2346547 1856326 + 8143674 2346547. 3. Simplify each of the following expressions, and factorize the result fully wherever possible. Show the steps of your argument clearly, and note any values of x that cannot apply. 3 2 6 (a) + 2 (b) x3 y 3 + x2 y x y 2 (c) x7 y 4 x4 y 7 2 x (x + 3) x 3x x 9 (d) x 2 x y y 2 4. Solve for the various values of x that satisfy each of these equations. 2 5 x 3 2x + 1 2 x 11 5 (a) = (b) = (c) +2= 3 4x 4 3x 4x 1 8x + 3 x 3 3 x 5. Suppose 40% of the dollar value of a stock portfolio consists of a high risk component, and 30% of the dollar value is in a safe category of investments. If $20,000 is added to portfolio in the high risk component, the safe component now represents just 20% of the value of the portfolio. What is the new total dollar value of the portfolio? Calculus 6. Let L be the line specified by 3x + 5y 75 = 0. (a) Find the points of intersection of this line with the x- and y-axes. (b) Find the distance between these intercepts. (c) Find the gradient of the line L. (d) Find the equation of the line perpendicular to L passing through the point (2, 1). 7. The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Assuming that the relationship between the cost and the number of chairs is linear, graph the cost against the number of chairs. Put cost on the vertical axis and the number of chairs on the horizontal. (b) Find an equation for the line. (c) What does the slope of the line mean in terms of costs and chairs? 8. The Paterson company manufactures banjos. The figure below shows Paterson's total production cost, C(q), as a function of the number, q, of banjos produced. The vertical axis gives cost in dollars. Thus, from the graph, one sees that the cost of producing 50 banjos is $9,000.00. The average cost, a(q), of producing a quantity, q, of banjos can be computed by taking the total cost, C(q), and dividing it by the quantity q. That is, a(q) = C(q)/q. (a) Explain the basic shape of the curve. What is the significance of the non zero vertical intercept? Why is it always increasing? Why is it initially increasing at a decreasing rate? Why does the curve then change to increasing at an increasing rate? C(q) (b) Complete the following table of values for the average cost function, a(q) = , and using this table q estimate the quantity of banjos which gives the minimum average cost per banjo. Sketch the graph of a(q) on a copy of the figure; choose an appropriate vertical scale. q a(q) 5 $566 10 20 30 $261 40 50 2 60 70 85 95 100 $170 (c) Place lines on the diagram on the diagram which have slopes equal to a(5), a(30), and a(100) and pass through the points (5, c(5)), (30, c(30)), and (100, c(100)) respectively. (d) Let q denote the quantity at which the average cost is minimized. Use the graph to estimate the value of q and the average cost at q , namely a(q ). (e) Place a line on the diagram which has slope equal to a(q ). C 18, 000 16, 000 14, 000 12, 000 10, 000 8, 000 6, 000 4, 000 2, 000 10 20 30 40 50 60 70 80 90 100 q 9. The value V (in thousands of dollars) of an antique car is given by the formula V = 40 (1.15)t , where t is the number of years since the beginning of the year 2015. (a) What is the value, to the nearest 100 dollars, of the car at the beginning of 2017? (b) At the beginning of 2017, estimate how fast the value of the car is changing. Give units in your answer. To compute this, use approximations based on the definition of the derivative and provide an answer with an accuracy of 2 significant figures. (To do this question, you don't need to have had a lecture on exponential functions, nor know the formula for the derivative of an exponential function. Indeed, the instructions in the question point you away from using any such knowledge or rules. You will use the formula that the rate of change of f (t) at t = a f (a + h) f (a) is given by f 0 (a) = lim if this limit exists.) h!0 h (c) Use technology to zoom in on the graph of V near the point t = 2 to check the answer from the previous part. (d) Using your result for value at the beginning of 2017, as well as the rate of change of the value at t = 2, estimate the value of the car in the middle of 2017. 3