Math 19: Calculus Winter 2008 Instructor: Jennifer KlokeQuiz 2 SolutionsMonday, January 281. State the definition of the derivative of f(x) at the p oint x = a. Specifically,Solution:f0(a) = limx→af(x) − f(a)x − a.orf0(a) = limh→0f(a + h) − f (a)h.2. Which of the following is not an expression for the s lope of the tangent line to the curvey = f(x) at the point P = (a, f(a)), (a 6= 0)?Solution: (b)limx→af(x) − f(a)x3. Find the equation of the tangent line to the curve y = x2+ 1 at the point (1, 2). Feel freeto use the fact that limx→1(x2+ 1) − 2x − 1= 2.Solution: The slope of the tangent line to the curve y = x2+ 1 = f (x) at the point (1, 2)is given by:limx→1f(x) − f(1)x − 1= limx→1(x2+ 1) − 2x − 1= 2as is given.Thus the equation is y − 2 = 2(x − 1).4. Suppose I need to know if there a root of f (x) = ln(x + 1) +√x − 1 between 0 and 2.What theorem can I use to find out if such a root exists? What are the two hypothesesof this theorem that I must check before I can apply it?Solution: I can use the Intermediate Value Theorem to solve this problem. The twohypotheses for this theorem are:(a) f(x) must be a continuous function on the interval [0,2].(b) The number 0 must be between f (0) and f(2).5. Let f(x) =0 : x < −21 : −2 ≤ x ≤ 01x−3: x > 0f(x) is continuous on which of the following intervals?Solution: (b)
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