Gain Mathematical in Calculus Through Multiple Representations!

Gain Mathematical in Calculus Through Multiple Representations!

Gain Mathematical

in Calculus through Multiple Representations!

Nancy Norem Powell

BloomingtonHigh School

1202 E. Locust

Bloomington, IL 61701

Email:

Website:

Nancy Powell’s

Favorite Graphing Activities

to Learn Calculus

Nancy Norem Powell, NBCT

Lead Teacher, Mathematics Department

BloomingtonHigh School

1202 E. Locust

Bloomington, IL 61701

Derivatives

Activity 1: (p. 202)

You are given the graph of a function on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the graph of the derivative of each function over the same interval.

Activity 2: Extrema(p. 273)

You are given the graphs of the derivatives of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). For each, indicate the locations of the local maxima and local minima (if any) of the original function.

Activity 3: Mean Value Theorem(p. 287)

For each of the function graphs shown, the grid lines are spaced 1 unit apart both horizontally and vertically. Approximate the slope of the secant line connecting the leftmost and rightmost points visible on the graph. If possible, locate at least one point at which the slope of the tangent line is the same as the slope of this secant line.

Activity 4: Concavity(p. 326)

You are given the graphs of the derivatives y = f’(x) of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). For each, indicate the locations of the points of inflection for the graphs of the original function y = f(x). Where is the graph concave up or concave down?

Culminating Activity5 – Derivatives:

The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs of f(x) that satisfy these requirements:

  1. For all f (x) >0, f ‘(x) > 0, f “(x) > 0
  2. For all f (x) >0, f ‘(x) > 0, f “(x) = 0
  3. For all f (x) >0, f ‘(x) > 0, f “(x) < 0
  4. For all f (x) > 0, f ‘(x) < 0, f “(x) > 0
  5. For all f (x) > 0, f ‘(x) < 0, f “(x) = 0
  6. For all f (x) > 0, f ‘(x) < 0, f “(x) < 0
  7. For all f (x) > 0, f ‘(x) = 0, f “(x) = 0
  8. For all f (x) < 0, f ‘(x) > 0, f “(x) > 0
  9. For all f (x) < 0, f ‘(x) > 0, f “(x) = 0
  10. For all f (x) < 0, f ‘(x) > 0, f “(x) < 0
  11. For all f (x) < 0, f ‘(x) < 0, f “(x) > 0
  12. For all f (x) < 0, f ‘(x) < 0, f “(x) = 0
  13. For all f(x) < 0, f ‘(x) < 0, f “(x) < 0
  14. For all f(x) < 0, f ‘(x) = 0, f “(x) = 0

Integrals

Activity 6: Area under a curve(p. 392)

You are given the graph of a function f on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of each of the following definite integrals by using x = 1 unit and counting “square units.”

If one or more of these definite integrals cannot be estimated for a particular graph, explain why.

Activity 7: Slope Fields(p. 402)

You are given the graph of a function y = f(x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the slope field for the antiderivatives of f. Then use this slopefield to sketch the graph of an antiderivative F of f over the interval [0,5], assuming F(1) = -2.

Activity 8: Integrals as Accumulated Area Functions(p. 426)

You are given the graph of a function y = f(x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, sketch the graph of the area function

Over the interval [1,5] for each function. Then sketch the graphs of the derivative of each area function and compare it with the original function’s graph.

Activity 9: Trapezoidal rule(p. 440)

You are given the graph of a function y = f(x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of

by using x = 1 unit and the trapezoidal rule.

Activity 10: Absolute Value functions and Integrals(p. 463)

You are given the graph of a function y = f(x) on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, estimate the value of

for each function. (If the graph does not appear over certain parts of the interval, assume that f(x) = 0 for these inputs.)

Activity 11: Tying things together–Derivatives and numerical approximation techniques: (p. 443)

The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy these requirements and determine:

a)Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) will always produce an underestimate for ?

b)Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) will always produce an overestimate for ?

c)Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) will always produce a exact answer for ?

d)Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) will there not be enough information to determine the relationship of the estimation to ?

  1. For all f (x) > 0, f ‘(x) > 0, f “(x) > 0
  2. For all f (x) > 0, f ‘(x) > 0, f “(x) = 0
  3. For all f (x) > 0, f ‘(x) > 0, f “(x) < 0
  4. For all f (x) > 0, f ‘(x) < 0, f “(x) > 0
  5. For all f (x) > 0, f ‘(x) < 0, f “(x) = 0
  6. For all f (x) > 0, f ‘(x) < 0, f “(x) < 0
  7. For all f (x) > 0, f ‘(x) = 0, f “(x) = 0
  8. For all f (x) < 0, f ‘(x) > 0, f “(x) > 0
  9. For all f (x) < 0, f ‘(x) > 0, f “(x) = 0
  10. For all f (x) < 0, f ‘(x) > 0, f “(x) < 0
  11. or all f (x) < 0, f ‘(x) < 0, f “(x) > 0
  12. For all f (x) < 0, f ‘(x) < 0, f “(x) = 0
  13. For all f (x) < 0, f ‘(x) < 0, f “(x) < 0
  14. For all f (x) < 0, f ‘(x) = 0, f “(x) = 0


Here’s an Extra Treat – I use these graphs all year long as we explore functions and what we can do to them!

Exploring functions with the graphs of y = f (x) and y = g (x)as given below:

Before beginning the activities below, find the equations of each piece of the piecewise linear functions above. Also, find the points of interest on each graph – the intersections of the pieces and the intercepts.

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Activity 12: (p. 82)

Given the graphs of the two functions f and g shown above, graph the indicated functions.

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

  1. f + g
  2. g – f
  3. 3f
  4. y = f(x2)
  5. y = f (2/x)
  6. y = f (x – 2)
  7. y = f (2x)
  8. y = f (x +2)
  9. y = f (x) + 3
  10. f / g
  11. f ○ g
  12. g ○ f
  13. g2
  14. gf

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Activity 13 (p.177)

Use the graphs of the two functions f and g shown above to answer the following questions.

  1. What is the slope of y = f (x) at x = - 4
  2. What is the slope of y = g(x) at x = 1?
  3. What is the slope of y = f (x)+g (x) at x = 4?
  4. What is the slope of y = f (x) -g (x) at x = 0?
  5. What is the slope of y = f (g (x)) at x = 3?
  6. What is the slope of y = g (f (x)) at x = -2?
  7. What is the slope of y = 2 f (x) at x = 2?
  8. What is the slope of y = g (x)/ 3 at x = 3?
  9. Graph f (x)g(x). Over what intervals is fg linear?
  10. Graph f (x) / g(x). Over what intervals is f / g linear?

Activity 14 (p. 237)

Using the graphs of the two functions f and g shown above, compute the derivatives indicated.

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

  1. F ‘ (4); F (x) = f (x) + g(x)
  2. F ‘ (4); F (x) = f (x) - g(x)
  3. F ‘ (1); F (x) = f (x) g(x)
  4. F ‘ (1); F (x) =
  5. F ‘ (3); F (x) = 2f (x) - 3g(x)
  6. F ‘ (-3); F (x) =
  7. F ‘ (-2); F (x) = g(x2)
  8. F ‘ (1); F (x) = g(2x – 3)
  9. F ‘ (-3); F (x) = f ( g (x))
  10. F ‘ (1); F (x) = g ( f (x))

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Activity 15 (p. 374)

Using the graphs of the two functions f and g shown above, find the areas described.

  1. Find the area of the region bounded by y = f (x), the x-axis, x = -1, and x = 4.
  2. Find the area of the region bounded by y = g (x), the x-axis, x = -5, and x = -2.
  3. Find the area of the region bounded by y = f (x), the x-axis, x = -1, and the y-axis.
  4. Find the area of the region bounded by y = g (x), and the x-axis.
  5. Find the area of the region bounded by y = f (x), y = g(x), and the y-axis.
  6. Find the area of the region bounded by y = f (x), y = g(x), and the x-axis.

Activity 16 (p. 394)

Using the graphs of the two functions f and g shown above, calculate the following definite integrals.

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007

Activity 17 (p.476)

Using formulas for the volumes of cones and cylinders, determine the volumes of the solids of revolution described given the graphs of the two functions f and g as shown above.

  1. Region between the graph of f and the x-axis between x = 0 and x = 5 rotated about the x-axis.
  2. Region between the graph of g and x-axis between x = 1 and x = 4 rotated about the x-axis.
  3. Region between the graph of f and the x-axis between x = -5 and x = 0 rotated about the y-axis.
  4. Region between the graph of g and the x-axis between x = -5 and x = -2 rotated about the y-axis.

Appendix A:

8 Graphs for Activities 1 - 11

1.

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2.

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3.

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4.

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5.

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6.

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7.

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8.

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Graphs for Activities 1 – 11

1. / / 2. /
3. / / 4. /
5. / / 6. /
7. / / 8. /

Appendix B:

Graphs for Activities 12 – 17

f(x) / g(x)

Nancy Norem Powell Section #871 NCTM – Atlanta, GA – March 24, 2007