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Section 4.7

Section 4.7. Optimization Problems. Example 8. A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. Section 4.8. Differentials.

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Section 4.7

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  1. Section 4.7 Optimization Problems

  2. Example 8 A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent.

  3. Section 4.8 Differentials

  4. Tangent Line Approximations • What is the equation of a line tangent to at the given point ? • This is called the tangent line approximation (or linear approximation) of at .

  5. Example 1 Find the equation of the tangent line to the graph of at the given point. Use this linear approximation to complete the table.

  6. Differentials • Using our tangent line approximation, , when is small we have that • is typically expressed as and is called the differential of . • is denoted and is called the differential of .

  7. Example 2 Use the info to evaluate and compareand .

  8. Example 3 Use the info to evaluate and compareand .

  9. Example 4 Find the differential .

  10. Example 5 Find the differential .

  11. Example 6 Find the differential .

  12. Example 7 Use differentials and the graph of to approximate and .

  13. Example 8 The measurement of the radius of the end of a log is found to be 16 inches, with a possible error of ¼ inch. Use differentials to approximate the possible propagated error in computing the area of the end of the edge.

  14. Brief Review of 4.1-4.6

  15. 4.1 (p. 209 #25) Locate the absolute extrema of the function on the closed interval.

  16. 4.1 (p. 209 #27) Locate the absolute extrema of the function on the closed interval.

  17. 4.2 (p. 216 #15) Determine if Rolle’s can be applied. If so, find the in such that .

  18. 4.2 (p. 216 #19) Determine if Rolle’s can be applied. If so, find the in such that .

  19. 4.2 (p. 217 #43) Determine if the MVT can be applied. If so, find the related -value.

  20. 4.2 Determine if the MVT can be applied. If so, find the related -value.

  21. 4.3 (p. 226 #41) Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

  22. 4.3 (p. 226 #25) Identify the intervals where the function is increasing or decreasing and locate all relative extrema.

  23. 4.4 (p. 235 #19) Find the points of inflection and discuss the concavity of the graph of the function.

  24. 4.4 (p. 235 #27) Find the points of inflection and discuss the concavity of the graph of the function.

  25. 4.4 (p. 235 #47) Find all relative extrema. Use the Second Derivative Test where applicable.

  26. 4.4 (p. 235 #55) Find all relative extrema. Use the Second Derivative Test where applicable.

  27. 4.5 Find the limit.

  28. 4.5 Find the limit.

  29. 4.5 Find the limit.

  30. 4.6 (p. 256 #29) -intercepts: -intercept: First derivative: Second derivative: End behavior: Critical numbers: Inflection pts.:

  31. 4.6 (p. 256 #6) -intercepts: -intercept: Asymptotes: First derivative: Second derivative: End behavior: Critical numbers: Inflection pts.:

  32. Questions??? Study hard and good luck!!!

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