1. If a line intersects a parabola at two points X and Y, there is ...

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1. If a line intersects a parabola at two points X and Y, there is an area A enclosed between the parabola and the line. Take the line which is tangent to the parabola and parallel to the line that connects X to Y. (This line exists by the Mean Value Theorem!) Call the point of tangency Z. There is a triangle B with vertices at X, Y, and Z. (See figure below) 6 5 4 -5 -4 -3 -2 1 2 3 4 5 (a) Consider the parabola f(x) = x2 + 5 and the points X = (-1,6) and Y = (5,30). i. Using an integral, find the area of A. Show your work and explain your reasoning. ii. What are the coordinates of the point Z? iii. What is the area of the triangle B? (Perhaps you want to find this area with integrals. Perhaps you know a formula for the area of such a triangle. Perhaps you should look up a formula on the internet.) (b) For any parabola, (the area of A) = 4/3(the area of B). Does your work in part (a) consistent with this fact? Explain. 2. For any real number b, define a function fb(x) = x3 - bx. For example: if b = 1, then fi(x) = x3 - x. Use technology to graph 1(x). If you're not sure about what b means, try graphing fo for a few different choices of b. The goal of this problem is to understand the shape of fb for all values of b. Using calculus, you'll be able to see all possible shapes. (a) Now study fo for all values of b. Using calculus and algebra³, find the critical numbers of fb. The critical numbers of fb depends on the value of b. Write down a formula in terms of b for all critical numbers of fb. If there are none, say SO. (Check your answer: For some values of b, there are no critical numbers. For one value of b, there is one critical number. For other values of b, there are two critical numbers.) (b) Continue to study fo for all values of b. Using calculus4, determine where fo is increasing and decreasing. The locations where fo is increasing and decreasing will depend on your value of b. (c) Again study fb for all values of b. Using calculus5, determine where fb is concave up and concave down. The locations where fo is concave up and concave down will depend on your value of b. (d) Continuing to study fb for all values of b, draw sketches of graphs of fb to show the different shapes fo could have. (There are 3.) Label each graph with the values of b that have the same shape for fb. (Check your work: For every real number b, have you described what the shape of fo looks like? If you graph some representative functions with your calculator, is the shape right?) TThis means don't use your calculator! 4 "This is what my calculator gave me" is not calculus. 5how how to use appropriate concepts from calculus here, not your calculator. 3. For all Z in the interval [0,5], the tangent line to the graph of f(x) = 49 - (x+2)² at Z, the x-axis, and the y-axis form a triangle. (a) For your favorite value of q in the interval [0,5], sketch the graph of f, the line tangent to f at q, the x-axis, and the y-axis. Shade in the triangle formed. (b) Define a new function A(z) on the interval [0, 5] that gives the area of the triangle as a function of Z. Write a formula for A(z). (c) Does A(z) have a maximum value on the interval [0, 5]? If so, determine the value of 2 at which the maximum occurs. If not, explain why not. (d) Does A(z) have a minimum value on the interval [0,5]? If so, determine the value of Z at which the minimum occurs. If not, explain why not. 4. Suppose that f(2) = 3 and f' (2) = 5. The domain f is the set of all real numbers. Suppose f'(x) > 0 and f"(x) < 0 on the entire domain of f. Explain your reasoning for each of the following. A good explanation would include words and may include a picture. (a) Does f have a value at X = 1.7? Explain how you know in at least one sentence. (b) Estimate f(1.7). (c) Is your estimate an overestimate, an underestimate, or can't you tell? (d) Suppose that f(-3) = 4. Is there a solution to f(x) = 0? Explain how you know in at least one sentence. (e) Estimate a solution to f(x) = 0. (f) Is your estimate an overestimate, an underestimate, or can't you tell? 5. Pick one of the following essay questions. Your answers should include sentences with proper punctua- tion. For the purpose of this exam, you should aim to answer this question well in 200-500 words. (a) Explain a concept that you saw in this course which was also present in another course that you were taking this semester. Offer details about the overlap in its uses and elaborate on the difference(s). Speculate on how this concept might be useful in subsequent courses (in any field). (b) Explain the notion of a "derivative". Give an example and explain its relevance in physical problems (as in physics for example). Optional additional question: If you had a function with a vector values or a function with vector inputs, how could you generalize the idea of derivative to this situation? (c) Compare and contrast the Intermediate Value Theorem and the Mean Value Theorem. In your discussion, include both the statements of the theorems and their applications. (d) Compare and contrast the first and second parts of the Fundamental Theorem of Calculus. In your discussion, include both the statements of the theorems and their applications. (e) Compare and contrast this course with your prevoius math courses. For example, what things did you see in pre-calculus that you most definitely needed in this course? What things from pre-calculus did not seem relevant for this course?

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