Math 234 Spring 2013 Exam 1 Version 1 Solutions

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1 Math 234 Spring 203 Exam Version Solutions Monday, February, 203 () Find (a) lim(x 2 3x 4)/(x 2 6) x 4 (b) lim x 3 5x x (c) lim x + (x2 3x + 2)/(4 3x 2 ) (a) Observe first that if we simply plug in x = 4, we end up with 0/0, so we need to begin by trying to factor and cancel like terms. We have x 2 3x 4 lim x 4 x 2 6 x 4 (x 4)(x + ) (x 4)(x + 4) x 4 x + x + 4 = = 5 8 (b) Since we are investigating the limit of a polynomial, we can simply plug in our value for x. We get lim x x3 5x = = = 0 (c) Here we are investigating a limit at infinity, so in order to solve we multiply top and bottom by to obtain: The highest power of x found in the denominator x 2 3x + 2 x 2 3x + 2 lim x + 4 3x 2 x + 4 3x x x 2 x x 2 = = 3 x 2 x 2

2 (2) Suppose that the function f(x) is defined by f(x) = { 5x if x < Ax 2 + 2x 3 if x. Find (a) lim f(x) x (b) lim f(x) x + (c) the value of A such that f(x) is continuous at x = (a) We have lim f(x) = 5() = 4 x (b) We have lim f(x) = A()2 + 2() 3 = A. x + (c) Observe that in order for f(x) to be continuous at x =, the limits from the right and left sides must be equal, and so we must have that 4 = A 3 = A. Furthermore, we must have that this limit is equal to the function value at x =. We see that the second function is the relevant one in evaluating f(). Since we are taking A = 3, we have f() = 3() 2 + 2() 3 = 4 x f(x). 2

3 (3) Find all points on the graph of f(x) = x + x 2 + x + where the tangent line is horizontal. Recall first that a horizontal tangent line is a tangent line whose slope is zero. Since we compute the slope of the tangent line by using the derivative, we first need to find the derivative of f(x). Since f(x) is a rational function, we will need the quotient rule in order to do so. We have f (x) = (x + ) (x 2 + x + ) (x + ) (x 2 + x + ) = (x2 + x + ) (x + ) (2x + ) = x2 + x + [2x 2 + 3x + ] = x2 + x + 2x 2 3x = x2 2x Next we need to figure out where this is zero. We have 0 = x2 2x 0 = x2 2x 0 = x(2 + x) 0 = x or 2 = x So the tangent is horizontal at (0, ) and ( 2, /3). 3

4 (4) (a) State the definition of f (x). (b) Use the the definition in (a) to compute f () when f(x) = x. (a) The derivative of the function f(x) with respect to x is the function f (x) given by f f(x + h) f(x) (x). h (b) To compute the derivative, we take f f(x + h) f(x) (x) h x+h x h x+h x x x x+h x+h x x+h x x+h h h x x + h h x x + h x x + h h x x + h x + x + h x + x + h x (x + h) h x x + h( x + x + h) h h x x + h( x + x + h) x x + h( x + x + h) = x x( x + x) = x 2 x = 2x x Evaluating this at x =, we have f () = 2 = 2 = 2 4

5 (5) It is estimated that t years from now, the population of a certain suburban community will be P (t) = 20 2 t + thousand people. (a) What will the population of the community be 3 years from now? (b) By how much will the population increase during the third year? (c) What happens to P (t) as t gets larger and larger? Interpret your result. (a) To find the population in 3 years, we plug t = 3 into our function to get P (3) = = 20 3 = 7. (b) To find out how much the population increases during the 3rd year, observe that the first year goes from t = 0 to t =, the second year from t = to t = 2, etc. So the 3rd year then goes from t = 2 to t = 3. We then need to find P (2) and subtract from P (3) to find the increase during that time period. We have P (2) = = 20 4 = 6 and so the increase during the 3rd year is P (3) P (2) = 7 6 =. So the population increases by 000 people during the 3rd year. (c) To see what happens as t gets larger and larger, observe that 2 t + > 0 will get smaller as t gets larger, until it ultimately ends up being zero. So we are subtracting smaller and smaller amounts from 20. So the population will increase at a diminishing rate, slowly approaching 20,000 inhabitants in the long term. 5

6 (6) A bookstore can obtain a certain book from the publisher at a cost of $3 per copy. The bookstore has been offering the book at the price of $5 per copy, and at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $ reduction in the price, 20 more books will be sold each month. Express the bookstore s monthly profit from the sale of this book as a function of the selling price. Since we want to write the store s monthly profit as a function of the selling price of the books, let x represent the price at which the books are being sold. We wish to compute the profit P (x), where Now P (x) = R(x) C(x). R(x) = (Price of book) (Number of books sold) = (x) ( (5 x)) and C(x) = (Cost of book) (Number of books bought) = (3) ( (5 x)) Observe that we assume that the bookstore will buy exactly as many books as they can sell. Observe also that the number of books that we calculate that they can sell depends on the price that they are charging. Check that it has the desired values at x = 5, 4, 3, and so on. Then P (x) = (x) ( (5 x)) (3) ( (5 x)) = (x 3)( (5 x)) 6