![]() ![]() Notice we are very roughly sketching this, as we don't have much information to work with. Now we can attempt to sketch the graph, starting at the point (0, 0). To start the sketch, we might note first the shapes we need: The derivative itself is not enough information to know where the function \(f\) starts, since there are a family of antiderivatives, but in this case we are given a specific point at which to start. Likewise, \(f\) should be concave up on the interval \((2, \infty)\). In the graph, \(f'\) is decreasing on the interval (0, 2), so \(f\) should be concave down on that interval. ![]() ![]() Likewise, \(f\) should be decreasing on the interval (1,3). In the graph shown, we can see the derivative is positive on the interval (0, 1) and \((3, \infty)\), so the graph of \(f\) should be increasing on those intervals. Recall from the last chapter the relationships between the function graph and the derivative graph: \( f(x) \) Use it to sketch a graph of \(f(x)\) that satisfies \(f(0) = 0\). The graph below shows \(f'(x)\) – the rate of change of \(f(x)\). Then we would look at the values of \(F\) at the endpoints to find which was the global min. It's the only critical point, so it must be a global max. \(f = F'\) goes from positive to negative there, so \(F\) has a local max at that point. Note that this is a different way to look at a problem we already knew how to solve – in Chapter 2, we would have found critical points of \(F\), where \(f = 0\): there's only one, when \(t = 3\). The maximum value is when \(t = 3\) the minimum value is when \(t = 0\). The area between \(t = 3\) and \(t = 4\) is much smaller than the positive area that accumulates between 0 and 3, so we know that \(F(4)\) must be larger than \(F(0)\). Since \( F(b) = F(a) \int\limits_a^b F'(x) \,dx \), we know that \(F\) is increasing as long as the area accumulating under \(F' = f\) is positive (until \(t = 3\)), and then decreases when the curve dips below the \(x\)-axis so that negative area starts accumulating. Where does \(F(t)\) have maximum and minimum values on the interval ? To learn more about the free Microsoft Word app, visit the Microsoft store.\) To learn more about the free Microsoft Word Viewer, visit the Microsoft Word website.ĭocx file: You need the Microsoft Word program, the Microsoft Word app, or a program that can import Word files in order to view this file. Download the free Adobe Acrobat Reader for PC or Macintosh.ĭoc file: You need the Microsoft Word program, a free Microsoft Word viewer, or a program that can import Word files in order to view this file. Pdf file: You need Adobe Acrobat Reader (version 7 or higher) to view this file. The video link did not work for the homework so we covered the video material in class. ![]() Lesson 7 - FTC with Graphs Notes (DOC 396 KB) Lessons 5-8 - HW from Book Key (5.4) (PDF 1.58 MB) Lesson 5 - FTC Evaluating Integrals (DOCX 142 KB)īasic Anti Derivatives Practice (PDF 11 KB)īasic Antiderivatives Worksheet Key (PDF 530 KB) Lesson 4 - HW from book Key (5.3) (PDF 516 KB) Lesson 4 - Average Value and Integral Rules (DOCX 186 KB) Lesson 3 - HW from book key (5.2) (PDF 491 KB) Lesson 3 - Book Pages for HW (PDF 421 KB) Lesson 3 - Riemann
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