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Understanding Balloon Volume Changes

Explore how a balloon's volume changes as it's filled with air using a given formula. Calculate average rate of change of radius and volume.

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Understanding Balloon Volume Changes

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  1. For Problems 1-2, we want to find how the volume, V, of a balloon changes as it is filled with air. We know V(r) = 4/3 πr3, where r is the radius in inches and V(r) is in cubic inches. The expression represents • The average rate of change of the radius with respect to the volume when the radius changes from 1 inch to 3 inches. • The average rate of change of the radius with respect to the volume when the volume changes from 1 cubic inch to 3 cubic inches. • The average rate of change of the volume with respect to the radius when the radius changes from 1 inch to 3 inches. • The average rate of change of the volume with respect to the radius when the volume changes from 1 cubic inch to 3 cubic inches. ConcepTest• Section 2.2 •Question 1

  2. ANSWER ConcepTest• Section 2.2 •Answer 1 (c) COMMENT: This is a nice way for students to see the formula and verbal description for average rate of change together.

  3. For Problems 1-2, we want to find how the volume, V, of a balloon changes as it is filled with air. We know V(r) = 4/3 πr3, where r is the radius in inches and V(r) is in cubic inches. Which of the following represents the rate at which the volume is changing when the radius is 1 inch? ConcepTest• Section 2.2 •Question 2

  4. ANSWER ConcepTest• Section 2.2 •Answer2 (c). Note that (d) would also be a reasonable answer because (a) and (b) provide approximations of the rate of change. COMMENT: Students should be aware that (b) is equivalent to

  5. For the function g(x) shown in Figure 2.3, arrange the following numbers in increasing order. (a) 0 (b) g’(-2) (c) g’(0) (d) g’(1) (e) g’(3) ConcepTest• Section 2.2 •Question 3

  6. ANSWER ConcepTest• Section 2.2 •Answer 3 (c), (d), (a), (b), (e) COMMENT: This can be used as an elimination question in a classroom quiz session.

  7. Which of the following expressions represents the slope of a line drawn between the two points marked in Figure 2.4? ConcepTest• Section 2.2 •Question 4

  8. ANSWER ConcepTest• Section 2.2 •Answer 4 (b) and (e). The coordinates of the two points shown are (a, F(a)) and (b, F(b)), so the slope of the line connecting them is COMMENT: You could repeat this question drawing the graph of a function that was increasing between a and b.

  9. Which of the following expressions represents the slope of a line drawn between the two points marked in Figure 2.5? ConcepTest• Section 2.2 •Question 5

  10. ANSWER ConcepTest• Section 2.2 •Answer 5 COMMENT: You could repeat this question drawing the graph of a function that was decreasing between x and x + ∆x.

  11. ConcepTest• Section 2.2 •Question 6 Let f(x) = x|x|. Then f(x) is differentiable at x = 0. (a) True (b) False

  12. ANSWER ConcepTest• Section 2.2 •Answer 6 COMMENT: Students often associate a minus sign with a number less than 0 rather than a number multiplied by – 1.

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