gain mathematical in calculus through multiple representations!

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Gain Mathematical Gain Mathematical in Calculus through in Calculus through Multiple Multiple Representations! Representations!

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Page 1: Gain Mathematical in Calculus through Multiple Representations!

Gain MathematicalGain Mathematical

in Calculus through in Calculus through

Multiple Representations!Multiple Representations!

Page 2: Gain Mathematical in Calculus through Multiple Representations!

to Learn Calculus!to Learn Calculus!

Nancy Norem PowellNancy Norem [email protected]@district87.org

Bloomington High SchoolBloomington High SchoolBloomington, ILBloomington, IL

Page 3: Gain Mathematical in Calculus through Multiple Representations!

Why use these activities?Why use these activities?

They help students They help students discoverdiscover and and visualizevisualize calculus concepts.calculus concepts.

They help students They help students exploreexplore and and connect connect calculus concepts.calculus concepts.

They They emphasizeemphasize and and focusfocus on conceptual on conceptual understanding of calculus by looking at understanding of calculus by looking at calculus graphically, numerically, and calculus graphically, numerically, and symbolically.symbolically.

Page 4: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity 1 Activity 1

You are given the graph of a function on a You are given the graph of a function on a grid. Assuming that the grid lines are grid. Assuming that the grid lines are

spaced 1 unit apart both horizontally and spaced 1 unit apart both horizontally and vertically, vertically,

sketch the graph of the derivative of each sketch the graph of the derivative of each function over the same interval.function over the same interval.

Page 5: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity 1 Activity 1 sketch the graph of the derivative of each function over the sketch the graph of the derivative of each function over the same intervalsame interval

4

2

-2

-5 5

m=-.8m=-.7

m=-.5m=-.3

m=0m=-.2 m=.2

m=.3m=.5

m=.7m=.8 2

-2

-5 5

Page 6: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 2 2 Extrema Extrema

You are given the graphs of the derivatives You are given the graphs of the derivatives of eight functions (grid lines are spaced 1 of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). unit apart both horizontally and vertically).

For each, indicate the locations of the local For each, indicate the locations of the local maxima and local minima (if any) of the maxima and local minima (if any) of the original function.original function.

Page 7: Gain Mathematical in Calculus through Multiple Representations!

4

2

-2

-4

-5 5

Derivatives:Derivatives: Activity Activity 2 2 Extrema: Extrema: You are given the graphs of the You are given the graphs of the derivatives of derivatives of f f

‘(x)‘(x).. For each, indicate the locations of the local maxima For each, indicate the locations of the local maxima and local minima (if any) of the original function.and local minima (if any) of the original function.

Positive Derivatives = Increasing Functions

Negative Derivatives = Decreasing Functions

Local MinLocal Min – f ‘ changes from decreasing to increasing and f ” is positive

Local MaxLocal Max– f ’changes from increasing to decreasing and f ” is negative

Page 8: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 3 3 Mean Value TheoremMean Value Theorem

For each of the function graphs shown, the grid For each of the function graphs shown, the grid lines are spaced 1 unit apart both horizontally lines are spaced 1 unit apart both horizontally and vertically. and vertically.

Approximate the slope of the secant line Approximate the slope of the secant line connecting the leftmost and rightmost points connecting the leftmost and rightmost points visible on the graph. If possible, locate at least visible on the graph. If possible, locate at least one point at which the slope of the tangent line is one point at which the slope of the tangent line is the same as the slope of this secant line.the same as the slope of this secant line.

Page 9: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 3 3 Mean Value TheoremMean Value Theorem Approximate the slope Approximate the slope

of the secant line connecting the leftmost and rightmost of the secant line connecting the leftmost and rightmost points visible on the graph.points visible on the graph.

4

2

-2

-4

-6

-5 5

If possible, locate at least If possible, locate at least one point at which the one point at which the slopeslope of the of the tangent linetangent line is the sameis the same as the slope as the slope of this of this secant linesecant line..

Slope is approx. 1/12

Tangent lineTangent line

Tangent lineTangent line

Secant lineSecant line

Page 10: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 4 4 ConcavityConcavityYou are given the graphs of the derivatives You are given the graphs of the derivatives y = f ’(x)y = f ’(x)

of eight functions (grid lines are spaced 1 unit of eight functions (grid lines are spaced 1 unit apart both horizontally and vertically). apart both horizontally and vertically).

For each, indicate the locations of the points of For each, indicate the locations of the points of inflection for the graphs of the original function inflection for the graphs of the original function y y = f (x).= f (x). Where is the graph concave up or Where is the graph concave up or concave down?concave down?

Page 11: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 4 4 Concavity Concavity You are given the graphs of the You are given the graphs of the derivatives derivatives y y = f ’(x). = f ’(x). For each, indicate the locations of theFor each, indicate the locations of the points of points of inflectioninflection for the graphs of the original function for the graphs of the original function y = f (x).y = f (x). Where is the graph concave up or concave down?Where is the graph concave up or concave down?

6

4

2

-2

-4

-6

-5 5

The slope of the first derivative is positive therefore the

second derivative is positive and the function is

CONCAVECONCAVEUPUP

CONCAVE CONCAVE DOWNDOWN

The slope of the first derivative is negative therefore the

second derivative is negative and the function is

Inflection Point

Page 12: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity Activity 5 5

1.1. For all For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 f (x) > 0, f ‘(x) > 0, f “(x) > 02.2. For all For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 f (x) > 0, f ‘(x) > 0, f “(x) = 03.3. For all For all f (x) > 0, f ‘(x) > 0, f “(x) < 0 f (x) > 0, f ‘(x) > 0, f “(x) < 0

4.4. For all For all f (x) > 0, f ‘(x) < 0, f “(x) > 0 f (x) > 0, f ‘(x) < 0, f “(x) > 05.5. For all For all f (x) > 0, f ‘(x) < 0, f “(x) = 0 f (x) > 0, f ‘(x) < 0, f “(x) = 06.6. For all For all f (x) > 0, f ‘(x) < 0, f “(x) < 0 f (x) > 0, f ‘(x) < 0, f “(x) < 0

7.7. For all For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 f (x) > 0, f ‘(x) = 0, f “(x) = 0

8.8. For all For all f (x) < 0, f ‘(x) > 0, f “(x) > 0 f (x) < 0, f ‘(x) > 0, f “(x) > 09.9. For all For all f (x) < 0, f ‘(x) > 0, f “(x) = 0 f (x) < 0, f ‘(x) > 0, f “(x) = 010.10. For all For all f (x) < 0, f ‘(x) > 0, f “(x) < 0 f (x) < 0, f ‘(x) > 0, f “(x) < 0

11.11. For all For all f (x) < 0, f ‘(x) < 0, f “(x) > 0 f (x) < 0, f ‘(x) < 0, f “(x) > 012.12. For all For all f (x) < 0, f ‘(x) < 0, f “(x) = 0 f (x) < 0, f ‘(x) < 0, f “(x) = 013.13. For all For all f (x) < 0, f ‘(x) < 0, f “(x) < 0 f (x) < 0, f ‘(x) < 0, f “(x) < 0

14.14. For all For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 f (x) < 0, f ‘(x) = 0, f “(x) = 0

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

Extension:

Using the clues on the right and your graphs, make a matching test for your calculus buddy!

Page 13: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity 5 - A Activity 5 - A

1.1. For all For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 f (x) > 0, f ‘(x) > 0, f “(x) > 02.2. For all For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 f (x) > 0, f ‘(x) > 0, f “(x) = 03.3. For all For all f (x) > 0, f ‘(x) > 0, f “(x) < 0 f (x) > 0, f ‘(x) > 0, f “(x) < 04.4. For all For all f (x) > 0, f ‘(x) < 0, f “(x) > 0 f (x) > 0, f ‘(x) < 0, f “(x) > 05.5. For all For all f (x) > 0, f ‘(x) < 0, f “(x) = 0 f (x) > 0, f ‘(x) < 0, f “(x) = 06.6. For all For all f (x) > 0, f ‘(x) < 0, f “(x) < 0 f (x) > 0, f ‘(x) < 0, f “(x) < 07.7. For all For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 f (x) > 0, f ‘(x) = 0, f “(x) = 0

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

Which description matches this graph?

f(x) has f(x) has positivepositive y- y-valuesvalues

The function The function is is increasingincreasing, , so f‘(x)so f‘(x) is is positivepositive!!

The function The function has has no no concavityconcavity – it – it is linear so is linear so f”(x)f”(x) = 0! = 0!

f(x)

xba

Page 14: Gain Mathematical in Calculus through Multiple Representations!

Derivatives:Derivatives: Activity 5 - Activity 5 - B B

1.1. For all For all f (x) < 0, f ‘(x) > 0, f “(x) > 0 f (x) < 0, f ‘(x) > 0, f “(x) > 02.2. For all For all f (x) < 0, f ‘(x) > 0, f “(x) = 0 f (x) < 0, f ‘(x) > 0, f “(x) = 03.3. For all For all f (x) < 0, f ‘(x) > 0, f “(x) < 0 f (x) < 0, f ‘(x) > 0, f “(x) < 04.4. For all For all f (x) < 0, f ‘(x) < 0, f “(x) > 0 f (x) < 0, f ‘(x) < 0, f “(x) > 05.5. For all For all f (x) < 0, f ‘(x) < 0, f “(x) = 0 f (x) < 0, f ‘(x) < 0, f “(x) = 06.6. For all For all f (x) < 0, f ‘(x) < 0, f “(x) < 0 f (x) < 0, f ‘(x) < 0, f “(x) < 07.7. For all For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 f (x) < 0, f ‘(x) = 0, f “(x) = 0

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

Which description matches this graph?

f(x)

xba

f(x)f(x) has has negative negative valuesvalues

f (x)f (x) is is decreasingdecreasing, so , so the the f ‘(x)f ‘(x) is is negativenegative ! !

f (x)f (x) is is cconcave downoncave down so f “(x) is so f “(x) is negativenegative!!

Page 15: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 6 Activity 6 Area under a curveArea under a curve You are given the graph of a function f on a grid. You are given the graph of a function f on a grid. Assuming that the grid lines are spaced 1 unit Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, apart both horizontally and vertically,

estimate the value of each of the following definite estimate the value of each of the following definite integrals by using integrals by using x = 1 unit and counting “square x = 1 unit and counting “square units.”units.”

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

5 3 2

1 3 6( ) ( ) ( )f x dx f x dx f x dx

Page 16: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 6- Activity 6-A A Area under a curveArea under a curve You are given the graph of a function You are given the graph of a function ff - - estimate the value of each of the estimate the value of each of the following definite integrals by using following definite integrals by using x = 1 unit and counting “square units.”x = 1 unit and counting “square units.”

6

4

2

-2

-4

-6

-5 5

IntervalInterval Approx.Approx. Area Area

1-21-2

2-32-3

3-43-4

4-54-5Total AreaTotal Area

5

1( )f x dx

- 4.5 sq u- 4.5 sq u- 2.5 sq u- 2.5 sq u- .75 sq u- .75 sq u

-7.5 sq u-7.5 sq u

.25 sq u.25 sq u

Page 17: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 6- Activity 6-B B Area under a curveArea under a curve

6

4

2

-2

-4

-6

-5 5

IntervalInterval Approx.Approx. Area Area

-3 to -2-3 to -2

-2 to -1-2 to -1

-1 to 0-1 to 0

0 to 10 to 1

1 to 21 to 2

2 to 32 to 3Total AreaTotal Area

- 5 sq u- 5 sq u- 4.5 sq u- 4.5 sq u- 2.5 sq u- 2.5 sq u

-17.0 sq u-17.0 sq u

3

3( )f x dx

.25 + - .25 = 0 sq u.25 + - .25 = 0 sq u0 sq u0 sq u

- 1.5 sq u- 1.5 sq u-3.5 sq u3.5 sq u

Page 18: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 6-C Activity 6-C

Area under a curveArea under a curve

6

4

2

-2

-4

-6

-5 5

IntervalInterval Approx.Approx. Area Area6 to 56 to 5

5 to 45 to 4

4 to 34 to 3

3 to 23 to 2

2 to 12 to 1

1 to 01 to 0

-1 to 0-1 to 0

-2 to -1-2 to -1

Total AreaTotal Area

-- - 5 sq u - 5 sq u-- - 4.5 sq u - 4.5 sq u

- - - 2.5 sq u- 2.5 sq u-- - .75 sq u - .75 sq u

16.7516.75 sq usq u

2

6( )f x dx

- - .75 sq u.75 sq u- - .25 sq u.25 sq u

-- - 3.5 sq u - 3.5 sq u - - .75 sq u.75 sq u

Page 19: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 7 Activity 7 Slope FieldsSlope Fields You are given the graph of a function You are given the graph of a function y = f (x)y = f (x) on a on a grid. Assuming that the grid lines are spaced 1 grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically, unit apart both horizontally and vertically,

sketch the slope field sketch the slope field for the antiderivatives of for the antiderivatives of ff. .

Then use this slope field to sketch the graph of an Then use this slope field to sketch the graph of an antiderivative F of f over the interval [0,5], antiderivative F of f over the interval [0,5], assuming assuming F (1) = -2.F (1) = -2.

Page 20: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity Activity 7 7 Slope Fields: Slope Fields:

a)a) sketch the slope field for the antiderivatives of sketch the slope field for the antiderivatives of ff. . b) b) Then use this slope field to sketch the graph of an antiderivative F of f Then use this slope field to sketch the graph of an antiderivative F of f over the interval [0,5], assuming F (1) = -2.over the interval [0,5], assuming F (1) = -2.

6

4

2

-2

-4

-6

5

f (x) F (x)4

2

-2

-4

5

(0,-1)

(3,0)

(2,-1/3)(4, 1/3)

(5, 2/3)

(1,-2/3)

xx mm

00

11

22

33

44

55

-1-1

-1/3-1/3

1/31/3

-2/3-2/3

2/32/3

00

Page 21: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 8 Activity 8 Integrals as Accumulated Area FunctionsIntegrals as Accumulated Area Functions

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

sketch the graph of the area functionsketch the graph of the area function

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

1( ) ( )

xA x f t dt

Page 22: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity Activity 8 8 Accumulated Area under a curveAccumulated Area under a curve

You are given the graph of a function You are given the graph of a function ff - - sketch the graph of the area function over the interval sketch the graph of the area function over the interval [1,5] for the function.[1,5] for the function.

IntervalIntervalApprox.Approx. AreaArea

AccumulatedAccumulated AreaArea

1 to 11 to 1

1 to 21 to 2

2 to 32 to 3

3 to 43 to 4

4 to 54 to 5

1( ) ( )

xA x f t dt

6

4

2

-2

-4

-6

5

- 1.0- 1.0

0.50.5

1.01.0

- 0.5- 0.5

- 1.0- 1.0

00

2.52.5

1.51.5

00

3.53.5

4

2

-2

5

Page 23: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 9 Activity 9 Trapezoidal rule Trapezoidal rule You are given the graph of a function y = f (x) on a You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are spaced 1 grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and vertically,unit apart both horizontally and vertically,

estimate the value ofestimate the value of

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

5

1( )f x dx

Page 24: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity Activity 9 9 Trapezoidal rule :Trapezoidal rule :estimate the value ofestimate the value of

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

5

1( )f x dx

6

4

2

-2

5

1 2( )2trapezoid

heightA base base

1

34

2

5

Trap #Trap # AreaArea

11

22

33

44

55Approx. Approx.

AreaArea

Note:

All of these trapezoids will have a height of 1 unit.

0.75 u2

0.5 u2

1.5 u2

1.75 u2

0.75 u2

0.75 u2

Page 25: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 10 Activity 10 Absolute Value functions and IntegralsAbsolute Value functions and IntegralsYou are given the graph of a function y = f (x) You are given the graph of a function y = f (x) on a grid. Assuming that the grid lines are on a grid. Assuming that the grid lines are spaced 1 unit apart both horizontally and spaced 1 unit apart both horizontally and vertically, vertically,

estimate the value of estimate the value of

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

6

6( )f x dx

Page 26: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity Activity 10 10 Absolute Value functions and IntegralsAbsolute Value functions and Integrals

Estimate the value of for each Estimate the value of for each function.function.

6 2

6( ) 15 f x dx u

6

6( )f x dx

6

4

2

-2

-4

-6

-5 5

Total Area

Vs.

Net Area

6 2

611( ) 4 7f x dx u

Page 27: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity 11 Activity 11 Derivatives and numerical approximation techniques:Derivatives and numerical approximation techniques:The behavior of a function f over the interval [a,b] is described in The behavior of a function f over the interval [a,b] is described in terms of its derivatives. Sketch graphs that satisfy the given terms of its derivatives. Sketch graphs that satisfy the given requirements and determine:requirements and determine:

Which numerical approximation technique(s) (left endpoint, right Which numerical approximation technique(s) (left endpoint, right endpoint, midpoint, trapezoidal, and Simpson’s rule) for endpoint, midpoint, trapezoidal, and Simpson’s rule) for

•will always produce an will always produce an underestimateunderestimate??•will always produce an will always produce an overestimateoverestimate??•will always produce a will always produce a exact answerexact answer??•will there will there not be enough informationnot be enough information to determine to determine

the the relationship of the estimation?relationship of the estimation?

( )b

af x dx

Tying things togetherTying things together

Page 28: Gain Mathematical in Calculus through Multiple Representations!

Integrals:Integrals: Activity Activity 11 11

1.1. For all For all f (x) > 0, f ‘(x) > 0, f “(x) > 0 f (x) > 0, f ‘(x) > 0, f “(x) > 02.2. For all For all f (x) > 0, f ‘(x) > 0, f “(x) = 0 f (x) > 0, f ‘(x) > 0, f “(x) = 03.3. For all For all f (x) > 0, f ‘(x) > 0, f “(x) < 0 f (x) > 0, f ‘(x) > 0, f “(x) < 0

4.4. For all For all f (x) > 0, f ‘(x) < 0, f “(x) > 0 f (x) > 0, f ‘(x) < 0, f “(x) > 05.5. For all For all f (x) > 0, f ‘(x) < 0, f “(x) = 0 f (x) > 0, f ‘(x) < 0, f “(x) = 06.6. For all For all f (x) > 0, f ‘(x) < 0, f “(x) < 0 f (x) > 0, f ‘(x) < 0, f “(x) < 0

7.7. For all For all f (x) > 0, f ‘(x) = 0, f “(x) = 0 f (x) > 0, f ‘(x) = 0, f “(x) = 0

8.8. For all For all f (x) < 0, f ‘(x) > 0, f “(x) > 0 f (x) < 0, f ‘(x) > 0, f “(x) > 09.9. For all For all f (x) < 0, f ‘(x) > 0, f “(x) = 0 f (x) < 0, f ‘(x) > 0, f “(x) = 010.10. For all For all f (x) < 0, f ‘(x) > 0, f “(x) < 0 f (x) < 0, f ‘(x) > 0, f “(x) < 0

11.11. For all For all f (x) < 0, f ‘(x) < 0, f “(x) > 0 f (x) < 0, f ‘(x) < 0, f “(x) > 012.12. For all For all f (x) < 0, f ‘(x) < 0, f “(x) = 0 f (x) < 0, f ‘(x) < 0, f “(x) = 013.13. For all For all f (x) < 0, f ‘(x) < 0, f “(x) < 0 f (x) < 0, f ‘(x) < 0, f “(x) < 0

14.14. For all For all f (x) < 0, f ‘(x) = 0, f “(x) = 0 f (x) < 0, f ‘(x) = 0, f “(x) = 0

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

Y-values are positive

Function is

•Increasing

•Concave up

Overestimate given by

Right Endpoint

Underestimate given by

Left endpoint

Exact Area given by

Unable to determine:

Midpoint

4

2

5 ba

Simpson’s

Trapezoid

None