Math 180 Practice Midterm #l l. A triangle has an unknown base b and height h. We know thta the hypotenuse is 30. Find a formula for the area in terms of the base I). 2. Find the domain of the function y = {(x) : Ix-+7: 3. a. Given for): J; and g(x)=167x2,tind fog(x) and go f(x) and their domains. b. If you want to nd the inverse function of f e g(x), how must you limit its domain so that the inverse function is truly a function? c. Find the inverse function of [a g(x) and its domain. 4. Given {(x) = ln(x+l), nd f'(x). 5 Graph y = log(x) 6. a. A patient is taking Immediate release Oxycontin, which is a powerful opiate pain reliever. The amount of Oxycontin in the bloodstream is modeled by an exponential decay function A : A02 '- where h = half life of the medicine. and An is the initial dose of medicine. The patient is given 10 mg of Oxycontin every 5 hours. Once someone takes 10 mg ofOxyeontin, assume they need 2 mg in Iheir blood to still get a good pain relief effect. if the half life of Oxycontin is 3.2 hours. how many hours after taking the rst dose will their blood level be down to a total of 2 mg, and thus they will be in pain again? b. Suppose the patient has been taking Oxycontin for many months and has built up tolerance. (Se, among other things, his liver is able to quickly metabolize the Oxycontin much faster than a person who never takes opiate pain relievers). For this patient. the half life is only 1.5 hauls. How many hours aer taking the rst dose will this patient have a blood level down to a total of 2 mg? e, For the patient in part b) above, this patient gets a second dose aer 6 houls. Will this patient be asking the nurse for more painkiller ahead of schedule? 7. Solve eh\" , 1400 : o 8. Graph y = tan(x). Graph y = arctan(x). 9. Find the limits below: . sin x 3. lim ( Ht )5 ) (note: this is to test that you have memorized what ling SM") (3 . H X Page > (1'12 ZOOM + Page 2 > of 12 ZOOM + b. lim tan(x) 2x2 +1 x 3 5x-1 x52 d. lim f(x) where f(x) = 0 x -+ 2 x =2 2x2 +1 x22 x+2 e. lim - * 7 2 x2 + 7x+10 f. lim x2 + 3x +1 x2 - 1 g. lim Vx+7+v2x-2 2x -18 h. lim In(x) x-+0+ i. lim ex j. lim - 1- -00 x2 + 1 k. lim - sin x+ 1) X 0of x lim x . 2 10. a. Use the Squeeze Theorem to prove that lim e & cos T - 0. b. Use the fact that, from geometry, cos(x) s sin(X)_ cos(x) and the Squeeze Theorem to prove that lim - sin(x) = 1. x 11. Using the 3 conditions which must be satisfied for continuity, show that f(x) is continuous at x = c. If it is not continuous, describe which type of discontinuity it is (i.e. removable, jump or infinite).Page 3 > of 12 ZOOM + x-4 a. f(x) = Vx-2 x *4 C = 4 2 x =4 b. f ( x ) = * + 3x -10 := 4. x2 - 9 c. f(x ) = *+2 C = 3 x - 3 x*0 d. f(x) = :=0 1 x=0 12. For what values of x is the following function continuous? f (x ) =_ sec (x Vx - 3 13. Find all the vertical and horizontal asymptotes and make a rough sketch of the graph of the function y = f(x) = 2x2 +1 x2 - 4x + 3 14. Find the limits below: a. lim x' +x+cos(x) 3x3 + 3 b. lim Vx2 + 5x+1 X - -00 x+2 15. a. Use the limit definition of a derivative to find the derivative of f(x) = 1. b. Use part a. above to find the equation of the tangent line to f(x) = _ at the point (1, 1). Graph f(x) = - and its tangent line. 16. a. Use the limit definition of a derivative to find the derivative of f(x) = x2 - 2x . b. Use part a. above to find the equation of the tangent line to f(x) = x2 - 2x at the point (1, -1). Graph f(x) = x -2x and its tangent line.Page of 12 I - ZOOM + 17. A baseball pitcher can throw a ball straight up at 120 feet per second straight up. Assume its height at time t is s(t) = -16t + 120t. What is its maximum height? 18. Given the graph of s(t) below, graph v(t) on a separate axis system. s(f) 1 2 3 4 5 6 t 19. Find all x values where f(x) is NOT differentiable. a. y = /x - 21 b. y = - x-1 20. Using basic differentiation rules, find f'(x) a. f (x ) = 4x2 + 3x 2 x 3 + 5 b . f ( x ) = Vx +-_ c. f (x) = 3x2ex 21. Find f"(x) : f(x) = 6x5