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MSU MTH 124 - MTH124 Practice Exam 2, Life Science, Key

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MTH 124 Practice Exam 2 Life Science Sections MTH 124Name: Section:About This Exam:This practice exam is designed to help you prepare for the upcoming exam. The typesof questions seen in this practice exam are representative of what you will see on theactual exam. However, there could be content on the actual exam that is notincluded in the practice exam.You will benefit the most from this exam by treating it like a real exam. You shouldstudy all the material before attempting the exam. When you do attempt the practiceexam allow yourself 50 minutes and only your calculator and a writing utensil.You will be expected to show all your work and provide justification where necessary onthe actual exam.Disclaimer: This practice test DOES NOT serve as an indication ofthe contents of the actual test. It only suggests a possible format.Page: 2 3 4 5 6 7 TotalPoints: 35 15 15 16 8 11 100Score:1MTH 124 Practice Exam 2 Life Science Sections MTH 124True or FalseAnswer the following questions by choosing true or false and clearly filling in exactly one choice. No credit will begiven for choices not clearly marked.1. (5 points) If x = c is a critical point of the function f(x), then f(c) must be a maximumor minimum. True√False2. (5 points) The rate of change of a function is always the largest at a point of inflection. True√False3. (5 points) If s(t) gives the position in feet t seconds after the start of an experiment, thens00(t) gives the acceleration.√True  False4. (5 points) The derivative of H(t) = 4t ln(t) is given by H0(t) = 41t. True√False5. (5 points) If a function is concave up, then that function must be increasing on its domain. True√False6. (5 points) Given the function v(s) we have that v0(7) = 0 and v00(7) = 200. This infor-mation means that at s = 7 the function v(s) has a minimum.√True  FalseMultiple ChoiceAnswer the following questions by choosing a single answer and clearly marking that choice. No credit will be givenfor choices not clearly marked.7. (5 points) Given that b(2) = 40, b0(2) = 6, and b00(x) < 0 which of the following values isnot a possible value of b(3)? b(3) = 30 b(3) = 35 b(3) = 40 b(3) = 45√b(3) = 502MTH 124 Practice Exam 2 Life Science Sections MTH 1248. Use the plot of Q00(x) on the domain [−4, 4] to answer the following questions.(a) (5 points) For x in [−4, 4], Q0(x) is increasing on which of the following?√[−4, −2) ∪ (2, 4) (−4, −2] ∪ [2, 4] (0, 3) [0, 3] x = 0 x = {−2, 2}(b) (5 points) Q(x) is concave up on which of the following?√[−4, −2) ∪ (2, 4) (−4, −2] ∪ [2, 4] (0, 3) [0, 3] x = 0 x = {−2, 2}9. (5 points) According to the first derivative test, if x = c is a critical point of a functionf(x) and f0(x) is negative to the left(x < c) of x = c and negative to the right(x > c),then which of the following must be true about f? f has a point of inflection at x = c. f has a maximum at x = c. f has a minimum at x = c.√f has a neither a maximum nor a minimum at x = c. None of the above answers are correct.3MTH 124 Practice Exam 2 Life Science Sections MTH 12410. (5 points) The derivative of N(t) =2t5− 4t(2.4)tis given by which of the following? N0(t) = (10t4− 4)(2.4)t+ (2t5− 4t) ln(2.4)(2.4)t N0(t) = (10t4− 4)(2.4)t− (2t5− 4t) ln(2.4)(2.4)t N0(t) =(10t4− 4)(2.4)t+ (2t5− 4t) ln(2.4)(2.4)t((2.4)t)2√N0(t) =(10t4− 4)(2.4)t− (2t5− 4t) ln(2.4)(2.4)t((2.4)t)2 N0(t) =10t4− 4ln(2.4)(2.4)tShort AnswerFor the following question show all your work. Make sure your answers include any relevant units. No credit willbe given for answers without the appropriate amount of work shown.11. A climatologist has determined the New York sea level in millimeters (mm) can be modeledby L(t) =4181 + 17.2e−0.04t, where t is time in years since 1990.(a) (6 points) How fast is sea level changing in the year 2015?Solution:L(t) =4181+17.2e−0.04t= 418(1 + 17.2e−0.04t)−1by exponent ruleL0(t) = −418(1 + 17.2e−0.04t)−2(17.2e−0.04t(−0.04)) by chain rule(twice)L0(25) = 1.970 . . .(b) (4 points) Explain the meaning of your answer from part (a) in the context of theproblem. Use any relevant units. If you couldn’t solve part (a) make up an answerto interpret.Solution: In the year 2015 the sea level is increasing by 1.970. . . millimeters per year.4MTH 124 Practice Exam 2 Life Science Sections MTH 12412. (8 points) Use derivative rules to determine the derivative of g(t) = (2t2− 5t)(e−0.15t) −1t3+ ln(t). Show all your work. You do not need to simplify.Solution:g0(t) = (2t2− 5t)0(e−0.15t) + (2t2− 5t)(e−0.15t)0+ t−4+1tProduct, power, ln ruleg0(t) = (4t − 5)(e−0.15t) + (2t2− 5t)(e−0.15t(−0.15)) + 3t−4+1tChain and power rule13. (8 points) An offshore oil well is leaking oil and creating a circular oil slick. If the radius,r(t), of the slick is growing at a rate of 2 miles/hour, find the rate at which the area isincreasing(mi2/hr) when the radius is 3 miles.(The area of a disc of radius r is A = πr2.)Solution:Givens : A = π(r(t))2, r(t) = 3, r0(t) = 2A0(t) = π2r(t)r0(t) chain ruleA0(t) = π2(3)2A0(t) = 12πmi2hourThus, at this time the area of the oil slick is increasing by 12π mi2every hour.5MTH 124 Practice Exam 2 Life Science Sections MTH 12414. Suppose g(x) = 2x3− 6x + 3.(a) (4 points) Use calculus to determine any critical points of g and list them below.Show your work.Solution:g0(x) = 6x2− 60 = 6x2− 6x = ±1 critical pointsg(x) has critical points as x = ±1.(b) (4 points) Use calculus to determine whether the critical points you found above areminima, maxima, or neither. Justify your answers.Solution:x < −1 -1 −1 < x < 1 1 x > 1g0(x) + 0 - 0 +g(x) % → & → %Thus by the first derivative test we see we have a maxima at x = −1 and a minima atx = 1. You could also use the second derivative test to justify your answer.6MTH 124 Practice Exam 2 Life Science Sections MTH 12415. A sheet of cardboard 3 ft. by 4 ft. will be made intoan open box by cutting equal-sized squares from eachcorner and folding up the four edges. A diagram of thebox is given to the right, the shaded region representsthe area that was cut from the corners of the box.(a) (5 points) Use the diagram above to determine expressions for the width and lengthof the box.Solution:3 − 2x4 − 2x(b) (6 points) Use your equations from part (a) to determine the dimensions of the boxthat would give the largest volume. The

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