Math 221 – Exam I – Friday February 20 – (50 Minutes)AnswersI. (25 points.) Define differentiable function, continuous function, andprove that a differentiable function is continuous. In your pro of you may use(without proof) the limit laws and high school algebra.Grader’s Comments: I assigned 5 points each for the definitions of “con-tinuous function” and “differentiable function”, and 15 points for the proofthat a differentiable function is continuous.Most students knew the definition of continuity at a point, namely f(a) =limx→af(x). Many did not remark that this must be true for all values of ain the domain of f , but I did not take off any points for that. Some insteadcharacterized continuity as being able to graph the function without liftingone’s pencil from the paper. This did not earn any points, as they were toldspecifically that this is NOT the definition.For the definition of a differentiable function, many simply wrote downthe definition of f0(a). As long as they did this correctly, I did not take offany points, but wrote a remark that the limit must exist and be finite for allvalues of a in the domain of f.For the proof, most students either managed to write down more or lesscorrectly what was in the handout, or wrote nothing close to a proof, so thegrading was close to binary. In some cases, the style was poor, or the logicwas a bit mangled, e.g., no ”=” signs where needed, missing limits, incorrectfirst line (f0(a) = 0 was common, followed by a more or less recitation of therest of the proof). I took off a few points in these cases.Some students, in lieu of a general proof, showed that some specific dif-ferentiable function is continuous. I did not give any points for this; afterall, the handout told them exactly what to write.II. (25 points.) Sketch the graph off(x) =−x if x < 0x if 0 ≤ x < 11 + x if x ≥ 1Then find each of the following or state that it does not exist.(a) limx→0f(x)(b) limx→1f(x)1(c) limx→2f(x)(d) limx→1−f(x)(e) limx→1+f(x)(f) f(1)Answer:III. (25 points.) Find each limit or state that it doesn’t exist. (Distinguishbetween an infinite limit and one which doesn’t exist.)(a) limx→∞x − 3√x2− 9.Answer:(b) limx→1x2+ x − 2x2− 1Answer:IV. (25 points.) (A) If f(3) = 7, f0(3) = 2 g(3) = 6, g0(3) = −10, find(a) (f · g)0(3)(b) (f + g)0(3)(c) (f/g)0(3)(B) If f(5) = 9, g(9) = 4, g0(9) = −5, f0(4) = 7, f0(5) = 11, and h(x) =g(f(x)), find h0(5).Answer:V. (25 points.) (A) If y =sin x + cos xsin xfinddydx. (Rememberdydxand Dxyare two notations for the same thing.)Answer: By the quotient ruledydx=(cos x − sin x) sin x − (sin x + cos x) cos xsin2x=sin2x − cos2xsin2x= −1sin2xGrader’s Comments: Many students simplified first as iny = 1 +cos xsin x2and then incorrectly asserted that sin x/ cos x is the tan x rather than cot x.Even if they did get y = 1 + cot x some forgot the negative sign indydx= −csc2x.(B) If f(x) = cosµ3x2x + 2¶, find f0(x).Answer: By the chain rule and the quotient rulef0(x) = −sinµ3x2x + 2¶ddxµ3x2x + 2¶= −sinµ3x2x + 2¶6x(x + 2) − 3x2(x + 2)2.Grader’s Comments: The most common mistake was to use the (incorrect)formula (g ◦ h)0(x) = g0(h0(x)) instead of the correct formula (g ◦ h)0(x) =g0(h(x))h0(x). These students wrote that the derivative is−sinµ6x(x + 2) − 3x2(x + 2)2¶.VI. (25 points.) Find the equation of the tangent line to the curvey = (x2+ 1)3(x4+ 1)2at the point (x, y) = (1, 32).Answer:---------------------------------------------------There are 208 scoresgrade range count percentA 135...150 29 13.9%AB 125...134 31 14.9%B 115...124 43 20.7%BC 100...114 52 25.0%C 85... 99 27 13.0%D 60... 84 18 8.7%F 0... 59 8 3.8%Mean score = 110.0. Mean grade =
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