# MATH120

MATH120 Midterm 2 August 9, 2021 Carefully show all your work. Justify all the steps in your answer with a sentence. Answers should be exact and may include fractions and nth roots. You are responsible to show your work clearly to the extent that other people can understand your solution. You must show all appropriate work in order to receive full credit for an answer. Please copy the pledge and sign. I pledge on my honor that I have not given or received any unauthorized assistance on this examination. 1. Let h(x) = (ex + 1) (x + 2)2 (x + 4)4 (x + 6)6 . (x − 2)2 (x − 4)4 (x − 6)6 A. (10 points) Differentiate y = h(x) using logarithmic differentiation. Do not need to simplify. B. (6 points) Find h0 (0). 2 2. (15 points) Sketch y = e−x . Indicate all the information from 6 categories. You must show all appropriate work to find all the information from 6 categories in order to receive full credit for your answer. 3. Given f (0) = 1, f 0 (0) = 2, f (1) = 2, f 0 (1) = 2, f (2) = 4, f 0 (2) = 1, g(0) = 2, g 0 (0) = 3, g(1) = 0, g 0 (1) = 2, g(2) = 0, g 0 (2) = −3, compute the following quantities: A. (4 points) d (f (x))3 dx ; x=0 B. (4 points) (f ◦ g ◦ f )0 (1); C. (4 points) d ln dx f (x) g(x) ; x=0 D. (4 points) E. (4 points) F. (4 points) d ln f (x) dx g(x) d f (eg(x) ) dx d f (x) dx g(x) ; x=0 ; x=2 . x=0 4. Determine whether the given statements are true and justify your answer. No point will be given to unjustified answer. A. (4 points) The functions y = ln x4 , y = 2 ln x2 , and y = 4 ln x are all distinct. B. (4 points) If 0 < b < 1, bx = logb x has a unique solution. C. (4 points) If α, β, b, and x are real numbers such that β 6= 0, b > 0, b 6= 1, x > 0, then logbβ xα = α logb x. β D. (4 points) For b > 0 and b 6= 1, if bα = logb α for some α, then α = bα . E. (4 points) There is no f (x) such that f 0 (x) = 1 1 + , f (0) = 0, and f (2) = 0. x−1 x+1 Bonus. (5 points) Find all the b’s such that b > 1 and bx = logb x has a unique solution. Z h i √ 3 e4x+1 − √ x + 2 5x − 3 dx. 5. (10 points) Find e 6. (10 points) Solve ln(2x − 2) + ln(x + 1) − ln x = ln 5 for x. 3 7. (10 points) Find the tangent line to y = ex ln x when x = e. Give an exact answer in simplest form. 8. (7 points) Francium-212(212 Fr) has a half-life of 20 minutes. If you begin with 20 grams, how much time passes until you have 4 grams left? Give an exact answer in simplest form. 9. (13 points) What is the rate of change of the volume of a spherical balloon with respect to time after 2 minutes from the initial time where the radius in inches after t minute(s) from the initial time is given by r(t) = t log3 (t + 1)? Give an exact answer in simplest form with the unit.

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