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Jeopardy

Jeopardy. Choose a category. You will be given the answer. You must give the correct question. Click to begin. Choose a point value. Choose a point value. Click here for Final Jeopardy. CONCAVITY AND INFLECTION. VOLUME AND CROSS SECTIONS. AVERAGE VALUE. LINEAR APPROXIMATIONS.

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Jeopardy

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  1. Jeopardy Choose a category. You will be given the answer. You must give the correct question. Click to begin.

  2. Choose a point value. Choose a point value. Click here for Final Jeopardy

  3. CONCAVITY AND INFLECTION VOLUME AND CROSS SECTIONS AVERAGE VALUE LINEAR APPROXIMATIONS Differential Equations 10 Point 10 Point 10 Point 10 Point 10 Point 20 Points 20 Points 20 Points 20 Points 20 Points 30 Points 30 Points 30 Points 30 Points 30 Points 40 Points 40 Points 40 Points 40 Points 40 Points 50 Points 50 Points 50 Points 50 Points 50 Points

  4. What is x=9?

  5. If the following is the graph of the derivative, f’(x), of a function f(x), what are the values of x at which f(x) has points of inflection, if f(x) is continuous and differentiable for all values x?

  6. x = -3, -1, and 1

  7. Dhaval and Oliver are arguing about the properties of a continuous and differentiable graph. Dhaval claims that if a graph is concave down, a line tangent to it will always be an over-estimate. Oliver refutes this, and adds that if a line is tangent to a graph to the left of its only point of inflection, and the 2nd derivative of the function changes from concave down to concave up at the point, then the 1st derivative will be negative. Dhaval disagrees with this, and claims, instead, that a graph must be concave up if, at a critical point, the first derivative is equal to zero. Before things get violent, Mr. Shay charges in on a unicorn to intervene. Who is right, and in what respects? Explain your answers.

  8. Dhaval’s first statement is correct. Oliver’s statement regarding a tangent line is incorrect, as a curve that is concave up will have a positive slope on at least its right half, so the slope of the tangent line must be positive. Dhaval’s second statement is also incorrect, since this only denotes a possible change in the sign of the slope of the graph, and not the concavity. Mr. Shay’s actions were correct, as he is Mr. Shay.

  9. Given the equation: • Which of the following are inflection points? • A) (0, -9001) only • B) (-4165, 4) only • C) (4, -4165) only • D) (0, -9001) and (4, 4165) only • E) None of the above

  10. C, (4,-4165)

  11. If the volume of a roly-poly is found by creating semi-circles, which are perpendicular to the x-axis and whose diameters lie on the region inside: • Then what is the area of the top of the shape that is formed when the roly-poly is squished to a cylinder of 0.1 in height, assuming that volume remains constant?

  12. A mushroom’s stem is calculated by rotating y = 1 around the x-axis from 0 to 4 and the cap is from 4 to 5 which is rotated about the x axis. Given the table below, use left Riemann sums to estimate the volume of the mushroom. A) 13π B) 12π C) 13 D) None of the above

  13. A; 13π

  14. A small bowl’s volume can be calculated by rotating the area between the following equations about the y axis, on x = [-1,1]; f(x) = – 1 • g(x) = x4 • What is the bowl’s volume?

  15. Region R is bounded by the graphs: • y=0 • x=2 • y= • What is the volume of the solid formed by regular hexagons, perpendicular to the x-axis, as its cross-sections?

  16. [Calculator Permitted] • Shay’s modern art sculpture is a solid created when the region bounded by the y axis, • y= cos(πx), y = 7-x, and y = (¼)x^2 –(7/4)x+7 when rotated around y = 8 • What is the volume of the solid?

  17. f(x) = 2x + 8 on [3, 10] Calculate, using a calculator, the average value of the function on [3, 10]

  18. The average value is 21.

  19. Using a midpoint sum with 2 subintervals, estimate the average value of f(x) on [1, 5]; A) 28 B) 7 C) 3 D) None of the above

  20. B; 7

  21. DAILY DOUBLE!!!

  22. DAILY DOUBLE: The function f(x) on [0, 6] is graphed below. Using both a left and right Riemann sum with 3 subintervals, estimate the average value of f(x) on [0,6]. If the equation of the graph is 12x2-3x+4, then what is the actual average value of f(x) on [0,6]?

  23. The estimated value is: The actual value is: 1

  24. If a freshman’s blood is being collected by Mr. Shay at a rate of 0.5L/hour at time = 4 hours, which is when he has lost 5 liters of blood, and there exists a value c in [1, 4] such that f’(c) = 1, what other conditions must be true to satisfy the mean value theorem, assuming that the freshman is losing blood at a continuous rate on t = [1, 4]?

  25. f(1) = 2L, and f(t) must be continuous and differentiable for all values t on [1,4]

  26. If the freshman mentioned in question 4 passes out when he loses 4L of blood, use the formula of the tangent line of the function at t = 4 to estimate the time t, in hours, when the freshman passes out. • Recall that f’(4) = 0.5L/h, and that f(4) = 5.

  27. The freshman passes out after two hours of bleeding. (t = 2)

  28. If the line tangent to f(x) at x = 9 is perpendicular to the graph of y = x + 278.59, and f(9) = 8, estimate, using the equation of the line tangent to f(x) at x = 9, estimate f(9.1).

  29. f(9.1) is roughly

  30. Given the graph below, approximate the value of f(x) (Blue function) at x = 1.2

  31. 3.4

  32. Given the following table of values, estimate f’(5) and use this to estimate f(5.3)

  33. (Calculator permitted): If the number of Mr. Shay jokes and references made in this class is represented by the graph • f(x) = 3, with x representing the number of projects assigned, take the derivative of f(x) to find the equation of the tangent line of f(x) when three projects have been assigned, then calculate the percent error of the approximation of f(3.1).

  34. 0.310%

  35. f’’(8) = g(4), 2(g’(x)) = f’(x), and g(x) = x2 – 6. • What is the estimated value of f’(8.1) as derived from f’’(x)? • A) 10 • B) 128 • C) 129 • D) None of the above

  36. C, 129

  37. Given the following slope field, represented by the equation,what is the equation of the line tangent to f(x) at (4,12)?

  38. y =

  39. Given the following differential equation • What is the function f(x) in terms of x, • if f(0) = 8?

  40. f(x) =

  41. Given the differential equation and f(1) = 9, what is the value of the constant of integration, c, if g(x) = h(4), and h(x) = 2?

  42. c = 79

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