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A critical point (or stationary point) of f(x) is a point (a;f(a)) such that f0(a) = 0. Recall that, geometrically, these are points on the graph of f(x) who have a \ at" tangent line, i.e. a constant tangent line. Critical Points f(x) Example 1: Find all critical points of f(x) = x3 3x2 9x+ 5. We see that the derivative is f0(x) = 3x2 6x 9.
Calculus II Lecture Notes, Baylor Jonathan Stan ll August 20, 2021 Text: Single Variable Calculus: Early Transcendentals, 4th Edition, Jon Rogawski and Colin Adams The course covers techniques of integration, applications of integration, and in nite series: Techniques of Integration Review: Substitution [Section 5.7] Integration by parts ...
This collection of solved problems covers elementary and intermediate calculus, and much of advanced calculus. We have aimed at presenting the broadest range of problems that you are likely to encounter—the
Next to some examples you’ll see [link to applet]. The link will take you to an online interactive applet to accompany the example - just like the ones used by your instructor in the lecture.
To check if a critical point is maximum, a minimum, or a saddle point, using only the first derivative, the best method is to look at a graph to determine the kind of critical point. For some applications we want to categorize the critical points symbolically.
Problem 11.1: Find all critical points for the following functions. If there are in nitely many, indicate their structure. For f(x) = cos(x) for example, the critical points can be written as ˇ=2 + kˇ, where kis an integer. a) f(x) = x6 3x2. b) f(x) = 4sin(ˇx) + 3 c) f(x) = exp( x2)x2. d) f(x) = sin(cos(ˇx))
A critical number (or critical value) is a number “c” that is in the domain of the function and either: Makes the derivative equal to zero: f′(c) = 0, or Results in an undefined derivative (i.e. it’s not differentiable at that place): f′(c) = undefined.
