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Calculus I

Calculus I. Section 4.7 Optimization Problems. Warm-up. Verify that the function satisfies the hypothesis of the Mean Value Theorem (MVT) on the given interval. Then find all values of c that satisfy the conclusion of the MVT. Mean Value Theorem.

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Calculus I

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  1. Calculus I Section 4.7 Optimization Problems

  2. Warm-up • Verify that the function satisfies the hypothesis of the Mean Value Theorem (MVT) on the given interval. Then find all values of c that satisfy the conclusion of the MVT.

  3. Mean Value Theorem • Let f be a function that satisfies the following hypotheses: • f is continuous on [a, b] • f is differentiable on (a, b) • Then there is a number c in (a, b) such that

  4. Optimization • One of the places calculus has wide application is in the field of optimization. From minimizing cost or maximizing profit in business to finding the minimum time for a dog to reach its ball, we can use the derivative to help us.

  5. Stewart’s Steps • Understand the problem • Draw a diagram • Introduce notation • Beat your head against the wall • Repeat step 4 • Use the material from sections 4.1-4.3 to solve.

  6. Important! • Do yourself a big favor and work through all of the examples in this section prior to starting the homework.

  7. Classic Example • A company wants to produce a flower box with an open top from a square piece of material, 3 ft wide, by cutting by cutting a square out of each of the 4 corners. Find the largest volume the box can have.

  8. Another Example • A company wants to produce a flower box with an open top and a square bottom that will hold 32,000 cm3 of soil. Find the dimensions that minimize the amount of material used.

  9. Same as Light? • Two vertical poles PQ and ST are secured by a rope from the top of the first pole to a point R on the ground between the poles and then the top of the second pole. Show that the shortest length of such a rope occurs when q1 = q2

  10. See Elvis Run • Please get out the “Do Dogs Know Calculus” handout. Verify

  11. See Elvis Run • Do the “Statistical analysis” of the data. We’ll run with the author’s assumption that the 4 points in the upper right are outliers and can be removed from the data and fit the line to the remaining data.

  12. Quiz on Friday • There will be an Optimath quiz on Friday.

  13. Another • Find the point on the line 6x + y = 9 that is closest to the point (–3, 1). • What is the equation of the line that passes through these 2 points?

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