Molly Leonard MA 131, section 601, Fall 2016 Instructor: Brenda Burns WebAssign Applications: Exponential, Logarithmic Functions (Homework) Current Score : 11 / 14 Due : Tuesday, December 13 2016 11:13 PM EST 1. 3/3 points | Previous Answers The population of a town is growing according to the differential equation The growth constant, k, is equal to 0.13 year -1. The size of the population at the start of the year 2000 was 15 thousand. Since the population is growing exponentially, the population in year t is given by Here, y is measured in thousands and t is measured in years since 2000. a. What is the population of the town at the start of the year 2005? (Enter your answer correct to one decimal place.) Population: 28.7 thousand. b. How many years does it take for the population to double? (Enter your answer correct to two decimal places.) Doubling time : 5.33 years. c. What will be the size of the population after three doubling times have passed? (That is, after three times the correct answer to part b have passed?) (Enter your answer correct to one decimal place.) Population after three doubling times : 120 thousand. 2. 5/5 points | Previous Answers The number of bacteria in a flask grows according to the differential equation In this question, time is measured in hours and the number of bacteria, y, is measured in millions. The number of bacteria at time t = 0 is 5 million. a. Enter a formula for the number of bacteria at time t y = 5e^(0.06t) b. What is the value of the growth constant? Growth constant : 0.06 per hour. c. How long does it take for the number of bacteria to double? (Enter your answer correct to two decimal places.) Doubling time : 11.55 hours. d. How many million bacteria will be present after 4 hours have passed? (Enter your answer correct to one decimal place.) Number present after 4 hours : 6.4 million. 3. 3/3 points | Previous Answers An isotope of a radioactive element has decay constant equal to 0.05 per year. Initially, there are 20 million atoms of the isotope present. Since the isotope decays exponentially, the number of atoms obeys the following equation In this question, time is measured in years and the number of atoms, P, is measured in millions. a. What is the value of the decay constant? Decay constant : 0.05 per year. b. What is the half-life of the isotope? (Enter your answer correct to two decimal places.) Half-life : 13.86 years. c. How many million atoms will be present after 6 years have passed? (Enter your answer correct to one decimal place.) Number present after 6 years : 14.8 million. 4. 0/3 points | Previous Answers An isotope of a radioactive element has half-life equal to 10 thousand years. Imagine a sample that is so old that most of its radioactive atoms have decayed, leaving just 6 percent of the initial quantity of the isotope remaining. How old is the sample? Give your answer in thousands of years, correct to one decimal place. Age : 40597.6 thousand years. WebAssign Exponential and Logarithmic Functions (Homework) Current Score : 21 / 30 Molly Leonard MA 131, section 601, Fall 2016 Instructor: Brenda Burns Due : Tuesday, December 13 2016 11:12 PM EST 1. 4/5 points | Previous Answers Solve each of the following equations for x. Either enter a symbolic expression (such as ln(3) - 2 or 4 exp(5) ) or the value of x as a number. Remember that e is entered as exp(1). If you enter a numerical value, be sure to enter at least four decimal places. If you need to take logarithms, WebAssign will expect you to use the natural logarithm. a. x = 1/8(ln(2/5)-5) b. x = sqrt(ln9) c. x = 4+e^2 d. x = e^(0.5) e. Where k is a constant. For this part of the question, you will have to enter your answer in symbolic form (i.e. in terms of k). Also, simplify your answer so that it involves the logarithm of a number that is greater than one. x = (1/k)ln(0.5) 2. 6/6 points | Previous Answers Differentiate the following functions Remember that e7x would be entered as exp(7x) in WebAssign. HINT: It might be possible to simplify one or more of these functions using the rules of exponentials before you differentiate. a. f '(x) = 2e^(2x) b. f '(x) = -(1/4)e^(-x/4) c. f '(x) = 8xe^(4x^(2)+4) d. f '(x) = e^(8x)(8x+1) e. f '(x) = e^(-2x^5)(-10x^8+4x^3) f. f '(x) = (-1)/(2sqrt(e^x)) 3. -/6 points Differentiate the following functions Remember that the natural logarithm of x would be entered as ln(x) in WebAssign. HINT: You may be able to simplify one or more of the following functions using the rules of logarithms BEFORE taking the derivative. a. f '(x) = b. f '(x) = c. f '(x) = d. f '(x) = e. f '(x) = f. f '(x) = 4. 11/13 points | Previous Answers Find the first and second derivative of the function where k is a non-zero constant. f '(x) = ke^(kx) f ''(x) = k^2e^(kx) a. Suppose that k is positive. Is the first derivative positive or negative? Negative Positive Is the second derivative positive or negative? Positive Negative Which of the following describes the graph of y=ekx? Decreasing and concave up Decreasing and concave down Increasing and concave up Increasing and concave down b. Suppose that k is negative. Is the first derivative positive or negative? Positive Negative Is the second derivative positive or negative? Positive Negative Which of the following describes the graph of y=ekx? Increasing and concave up Increasing and concave down Decreasing and concave down Decreasing and concave up Find the first and second derivative of the function for x greater than zero. f '(x) = 1/x f ''(x) = -(1/(x^2)) Is the first derivative positive or negative? Positive Negative Is the second derivative positive or negative? Negative Positive Which of the following describes the graph of y = ln x? Decreasing and concave up Decreasing and concave down Increasing and concave down Increasing and concave up Molly Leonard MA 131, section 601, Fall 2016 Instructor: Brenda Burns WebAssign Further Integration (Homework) Current Score : 21 / 40 Due : Tuesday, December 13 2016 11:01 PM EST 1. 5/5 points | Previous Answers In this question, you will use a substitution to carry out the following integration: If the answer requires a constant of integration, enter it as c. a. The integral involves the composite function . What u-substitution will simplify this term? u = g(x) = x^5+1 b. Find the derivative : = 5x^4 c. Transform the original integral into one involving u by using the substitution rule: Replace all occurrences of Replace by = in the integral by ( equivalently: replace . by ). u^6 (you do not need to enter du in your answer) d. Carry out the integration, and find the most general antiderivative (in terms of u). antiderivative : (u^7/7)+c e. Finally, rewrite your answer in terms of x by replacing u by g(x). = (1/7)(x^5+1)^7+c 2. 5/12 points | Previous Answers Integrate each of the following functions using substitution, finding the most general antiderivative. Also enter u, the function of x that you substitute. If your answer requires a constant of integration, enter it as c. a. = 2/3u^(3/2) u = x^3+1 b. = 2sqrt(x^5+3x+3) u = x^5+3x+3 c. = u = ln(x) d. = u = 2x^4+5 e. = 2sqrt(6x^6+3) u = 6x^6+3 f. = u= 3. 7/19 points | Previous Answers In this question, you will estimate the value of the integral using three different approximations. a. Subdivide the interval [1,4] into three sub-intervals of equal width and complete the following: = 1 a0 = 1 f(a0) = 0.6065 a1 = 2 f(a1) = a2 = 3 f(a2) = 0.66939 a3 = 4 f(a3) = x1 = 2 f(x1) = x2 = 4 f(x2) = x3 = 6 f(x3) = b. Calculate the approximate value of the integral using the trapezoidal rule. c. Calculate the approximate value of the integral using the midpoint rule. d. Calculate the approximate value of the integral using Simpson's rule. e. It is possible to show that an antiderivative of x e-x/2 is Using this antiderivative, calculate the exact value of the integral. Integral = 4. 4/4 points | Previous Answers In this question, you will investigate whether the improper integral converges or diverges. If it converges, you will find its value. a. Calculate the value of the integral where b is a finite number whose value is greater than one. Value = (b-1)/b b. Does the value of the integral approach a limit as b tends to infinity? If so, enter this limiting value: = 1