10/06 Practice MidtermSupplemental InstructionIowa State UniversityLeader: Matt ECourse: Math 165 Instructor: KramerDate: 10/06/10This is the first part of the test. No calculators are allowed. You will probably have only 30-40 minutes to finish this on the actual Midterm. Evaluate the following limits or state that they do not exist: 1) limn →∞3 n2−2 n+66 n2+32) limz → 12 z2+5 z−7z2−13) x → 1−¿−x3+6 x +2x2−1lim¿¿4) limx →(−1)−x3+6 x+2x2−15) limθ → 0sin(3 θ)∙cos (−2θ)tanθ6) Differentiate using the limit definition, NOT the derivative rules (power rule, chain rule): y=√x Differentiate using the derivative rules. You don’t have to simplify this one.Supplemental Instruction1060 Hixson-Lied Student Success Center v 294-6624 v www.si.iastate.edu7)y=5 sin(x3)√x This is still the first part of the test. Do not use a calculator. 8) Find dydx :x y3+2010= 2 y −4 csc x 9) Find: Dx97sin x10) Find: Dx6[50 x5−18 x4+4 x3−74 x2+201]This concludes the first part of the test. You may now begin using a calculator (but you can’t go back to questions 1 through 10 because you already handed those in). You must show all work. No credit is allowed for mere answers with no work shown. Justify any conclusions.11) f(x)=−6x3−3Find the equation of the tangent line to f(x) at x=112) Sketch and label a graph with all of the following properties:the graph is of a function, f(x)f (x) is continuous for all x except -1. There is a removable discontinuity at -1 f'(2)=DNE f(−2)=f(2)=3 limx →−1f(x)=1 limx → ∞f(x)=0 13) A 10 ft ladder, with one end on the ground, is falling over. Consider the angle the ladder makes with the ground. When the falling end is 6 ft above the ground, this angle is shrinking at a rate of π2radians per second. How fast is the end of the ladder falling at this moment?14) Consider the function f(x)=23x3−6 x2−8On what intervals (closed) is this function increasing? Decreasing?On what intervals (open) is this function concave up? Concave down?*BONUS PROBLEMS* -Your real midterm will not have any bonus problems. I just thought these would be good practice. 15) Assume the function in problem 14 gives the position (f) of an object along an axis as a function of time (x). Position is measured in meters. Time is measured in seconds. Take the rightside of this axis to be positive and the left side to be negative. Find the instantaneous velocity when x = 4 seconds. (Include sign and units)Find all time intervals (open) for which the object is speeding up (in either direction).16) A spherical bubble is growing at a rate of 27 cubic inches per second. How fast is its surface area increasing when its volume is 32 π3 cubic inches? (Surface area of a sphere = 4 π r2 )Answers1) 122) 923) +∞4) DNE (Does not exist)5) 36) 12√x (I used the h definition – I think it’s easier. I had to multiply by the conjugate to solve this problem. This might not be on the Midterm, but it was on Kramer’s last test, so I wanted to include it.)7) 5 ∙ cos(x3)∙3 ∙ x2∙√x−5 ∙ sin(x3)∙12x−12x (not simplified)8) x¿¿cot(x)– y34 csc¿dydx=¿9) cos⁡x10) 011)y−3=92( x−1)12) Many possible answers. All should includecorner at (2,3) [A vertical tangent line is okay instead of a corner]hole at (-1,1)horizontal asymptote y=013) 4 π ft/s or ≈ 12.57 ft / s14) Increasing: (−∞, 0]∪¿ Decreasing: [0,6 ] Concave Up: (3, ∞) Concave Down: (−∞ , 3)15) -16 m/s (0,3) and (6, ∞)16) 27 i