Coursehero >> Colorado >> Colorado >> MATH 1081

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

2 EXAM Section xxx October 7, 2005 NAME: Make sure to read the question carefully and answer the question that is asked. Show ALL relevant work so that partial credit may be given and indicate where the solution is. Lack of sufficient work may result in a loss of credit, even if a correct answer is given. Good luck!! "On my honor, as a University of Colorado at Boulder student, I have neither given nor received unauthorized assistance on this work." YOUR SIGNATURE: 1. Marginal Functions The cost (in dollars) to produce x color printers is given by C(x) = 400x + 100000. The marketing department has also established that the demand for these printers is given by p = -0.05x + 700, where p denotes the printer's unit price (in dollars) and x denotes the quantity demanded. (a) (3 points) What is the revenue function R(x)? Solution: The revenue can be found by multiplying price p by quantity x. So, R(x) = px = (-0.05x + 700)x = -0.05x2 + 700x. (b) (7 points) Find the marginal profit function and verify that it is equal to zero when x = 3000. Solution: Profit is "revenue minus cost" and "marginal" indicates the derivative. P (x) = R(x) - C(x) = (-0.05x2 + 700x) - (400x + 100000) = -0.05x2 + 300x - 100000 Taking the derivative, we find the marginal profit function to be P (x) = -0.1x + 300. So, when x = 3000, we see that the marginal profit function is P (3000) = -0.1(3000) + 300 = -300 + 300 = 0. 1 2. Rules of Differentiation (2 points each) Find the derivative of the function. You DO NOT need to SIMPLIFY your answer. (Note that the solutions have some simplifications, but these would not be required on this exam.) 1 3 (a) f (x) = 4x5 - x2 - 90 + x 5 2 Solution: f (x) = 4 5x4 - 7 1 (b) g(x) = - 5 3 x x Solution: 1 4 3 1 4 2x - 0 + x- 5 = 20x4 - 3x + x- 5 . 2 5 5 First, rewrite g(x) as a sum/difference of powers of x. g(x) = x- 3 - 7x-5 . 35 1 1 4 Then, g (x) = - x- 3 - 7 (-5)x-6 = - 4 + 6 3 x 3x 3 (c) h(x) = 3e-2x Solution: h (x) = 3 e-2x (-2) = -6e-2x (d) k(x) = (3x2 - 2x + 1)(4x3 + x2 - 4) Solution: Use the product rule (or if you want, "FOIL" the product first). k (x) = (3x2 - 2x + 1)(4 3x2 + 2x) + (3 2x - 2)(4x3 + x2 - 4) = (3x2 - 2x + 1)(12x2 + 2x) + (6x - 2)(4x3 + x2 - 4) 3x3 - 15x + 20 (e) R(x) = x Solution: First simplify the fraction to a sum/difference of powers of x by dividing each term of the numerator by the denominator x (or if you want, use the quotient rule). So, R(x) = 3x2 - 15 + 20x-1 . 20 Then, R (x) = 3 2x - 0 + 20 (-1)x-2 = 6x - 2 . x 1 2 3. Rules of Differentiation (4 points each) Find the derivative of the indicated function. You DO NOT need to SIMPLIFY your answer. (a) F (x) = (4x3 - 5x + 1)(9x2 - 4) 2 Solution: Product rule first, with chain rule applied when taking the derivative of the second term. 1 3 3 3 F (x) = (4 3x2 - 5)(9x2 - 4) 2 + - (4x3 5x + 1) (9x2 - 4) 2 (9 2x) = (12x2 - 5)(9x2 - 4) 2 + 2 1 3 (4x3 - 5x + 1) (9x2 - 4) 2 18x 2 More simplification is certainly possible, but not required 2x2 -1 x+1 3 (b) G(x) = e Solution: Exponential function chain rule with the quotient rule applied when taking the derivative of the inside function. 2x2 -1 2x2 -1 (x + 1)(2 2x) - (2x2 - 1)(1) 2x2 + 4x + 1 = e x+1 G (x) = e x+1 2 (x + 1) (x + 1)2 (c) H(x) = e3x + ln(4x) Solution: Generalized power rule (chain rule) with exponential function chain rule and logarithmic function chain rule applied when taking the derivative of the inside function. 1 1 1 3x 1 3x -1 -1 e + ln(4x) 2 e3x 3 + e + ln(4x) 2 3e3x + 4 = H (x) = 2 4x 2 x 5 (d) C(x) = Solution: 7x4 - 3x - 1 3x3 1 term. This is equivalent to 3 x-3 and NOT 3x-3 !!! 4 1 1 1 C (x) = 5 7x4 - 3x - 3 (74x3 -3- (-3)x-4 ) = 5 7x4 - 3x - 3 3x 3 3x Be careful with the 1 3x3 4 (28x3 -3+x-4 ) (e) P (x) = e3x ln(6x2 - 1) Solution: Quotient rule with exponential function chain rule and logarithmic function chain rule applied when taking the dervatives of the numerator and denominator, respectively. 2 2 2 2 12x 1 ln(6x2 - 1) 6xe3x - e3x 6x2 -1 [ln(6x2 - 1) e3x 6x] - [e3x 6x2 -1 12x] = P (x) = (ln(6x2 - 1))2 (ln(6x2 - 1))2 2 3 4. Tangent Line (5 points) Find the equation of the tangent line to the curve R(x) = the point x = 1. Solution: x2 + 7x + 1 at To find the equation of the tangent line, we need a point and the slope. Since we are looking at x = 1, the y-coordinate of the point of interest is found by plugging this into the original function. So, R(1) = (1)2 + 7(1) + 1 = 9 = 3 is the y-coordinate. To find the slope of the tangent line, we need the derivative of the function. R (x) = 1 2x + 7 1 2 (x + 7x + 1)- 2 (2x + 7) = 2 2 x2 + 7x + 1 So, at x = 1, the slope of the tangent line is R (1) = 2(1) + 7 9 3 9 = = = 6 2 2 9 12 + 7(1) + 1 2 3 Using the point-slope form of a line, an equation of the tangent line is y - 3 = (x - 1). (Note that 2 3 3 this is equivalent to y = x + . 2 2 4 5. Derivatives and a Graph (5 points) Use the graph of the function f (x) to answer the questions below. y-axis graph of y = f(x) x-axis a b c d e f g (a) Find the intervals where the first derivative f (x) > 0. Solution: This means the graph is increasing, so (a, b) and (d, f ). (b) Find the intervals where the second derivative f (x) < 0. Solution: This means the graph is concave downward, so (a, c) and (e, g). (c) How many solutions are there to the equation f (x) = 0? Solution: 3, because this means the tangent line is horizontal at that point, so at x = b, x = d, and x = f . (d) List all inflection points of the graph of the function. Solution: The inflection points are where the concavity changes, so at x = c and x = e. 5

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: