Learning Outcome: Sketch the graph of a function using extreme value theorem, mean valuetheorem, first derivative test, concavity and points of inflection (i.e. second derivative test) andlimits at infinity.Lesson: Extreme Value Theorem, Mean Value Theorem, First Derivative Test, Concavity andPoints of Inflection (i.e. second derivative test), Limits at Infinity and Curve SketchingGeneral Instructions: Consider the given steps to sketch the graph of the functions.Functions to be graphed:1. y =2×272-12. f (x) =3. f(x) = 2cosx + sin2x4. f (x)=5. y = 1 – 3x + 5×2 – x3Steps:for which f(x) is defined.1. Domain. The first step is to determine the domain D of f, that is, the set of values of x2. Intercepts. The y-intercept is f(0) and tells us where the curve intersects the y-axis. Tofind the x-intercepts, we set y = 0 ad solve for x. (If this is not easily done, the intercepts could beestimated.)3. Symmetry.(i) If f(-x) = f(x) for all x in D, that is the equation of the curve is unchanged when xis replaced by -x, then f is an even function and the curve is symmetric about the y-axis. Thismeans that our work is cut in half. If we know what the curve looks like for x20, then we need onlyreflect about the y-axis to obtain the complete curve. Some examples are y = x, y = x*, y =Ixl and y = cosx.(ii) If f(-x) = -f(x) for all x in D, then f is an odd function and the curve is symmetricabout the origin. Again we can obtain the complete curve if we know what it looks like for x20.Some simple examples of odd function are y = x, y = x , y = x and y = sinx.(i) If (x+p) = f(x) for all x in D, where p is a positive constant, then f is called aperiodic function and the smallest such number p is called the period. For instance, y=sin x hasa period 2/ and y=tan x has a period . If we know what the graph looks like in an interval oflength p, then we can use translation to sketch the entire graph.4. Asymptotes.(i) Horizontal Asymptotes. Either lim f(x) = Lor lim f(x) = L then the lineX-4 -00y=L is a horizontal asymptote of the curve y = f(x). If it turns out that lim f(x) = co (or – co), thenX-+00we do not have an asymptote to the right, but that is still useful information for sketching the curve.(ii) Vertical Asymptote. The line x = a is a vertical asymptote if at least one of thefollowing statements is true:1. lim f(x) = 00x -at2. lim f (x) = 003. lim f (x) = -cox-at4. lim f (x) = -cox-at(For rational functions you can locate the vertical asymptotes by equating the denominator to 0after canceling any common factors. But for other functions this method does not apply.)Furthermore, in sketching the curve, it is very useful to know exactly which of the statementsmentioned is true. If f(a) is not defined but a is an endpoint of the domain of f, then you shouldcompute lim f(x) or lim, f(x), whether or not this limit is infinite.x-at5. Intervals of Increase of Decrease. Use the Test for Monotonic Functions. Computef'(x) and find the intervals on which f'(x) is positive (f is increasing) and the intervals on whichf'(x) is negative (f is decreasing).6. Local Maximum and Minimum Values. Find the critical numbers of f (the numbers cwhere f'(c) = 0 or f'(c) does not exist.). The use the First Derivative Test. If f’ changes frompositive to negative at a critical number c, then f(c) is a local maximum. If f’ changes from negativeto positive at c, then f(c) is a local minimum. Although it is usually preferable to use the FirstDerivative Test, you can use the Second Derivative Test if c is a critical number such that f”(c) #0. Thenf”(c) > 0 implies that f(c) is a local minimum, whereas f”(c) < 0 implies that f(c) is alocal maximum.7. Concavity and Points of Inflection. Compute f”(x) and use the Test for Concavity. Thecurve is concave upward where f”(x) > 0 and concave downward where f” < 0. Inflection pointsoccur where the direction of concavity changes.8. Sketch the Curve. Using the information in items 1-7, draw the graph. Draw in theasymptotes as broken line. Plot the intercepts, maximum and minimum points and inflectionpoints. Then make the curve pass through these points, rising and falling according to item 5, withconcavity according to item 7 and approaching the asymptotes. If additional accuracy is desirednear any point, you can compute the value of the derivative there. The tangent indicates thedirection in which the curve proceeds.
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