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Date_____________ Name___________________________________________

Work Smarter Not Harder Lab 5 Gough and Gough

27

Exploring the Definition of the Derivative Lab 5

1. Open the document Lab 05 Definition of Derivative h approaches 0. Press / tomove to page 1.2.

2. Find the value of the derivative of f(x) =x2 +1at x = 2 . Investigate graphically,numerically, and analytically. Press / to move to page 1.3.

Graphically

3. Page 1.3 shows the graph of f(x) =x2 +1, the point , the value of , and a point units horizontally away from . Plot the point which is horizontally

h = 2 units away from on the graph below, sketch in the line passing through

the two points, and compute the slope of the line in the space to the right of the graph.

4. The definition of the derivative states that approaches . In the step above, h = 2 .Double click on the value of and change it so that h = 1. As in the step above, plot

and compute using and the new point where h = 1.

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5. Let . As in the step above, plot and compute using and the new pointwhere .

6. As gets closer and closer to , the slope of the secant line is approaching _______.7. As approaches , the secant line approaches the _______________________ line.

Numerically

8. Page 1.4 is a Calculator page. The slope of the secant line passing through andthe point where h = 2 using function notation has been calculated. Record the slopein the table below.

9. Does this calculation agree with your calculation in question 3? _________________

f(x + h) f(x)

h

f(x + h) f(x)

h

2 -2

1 -1

0.5 -0.5

0.1 -0.1

0.01 -0.01

10.Repeat the calculation for each value of in the table above.11.As the -values approach from the right, the slope is approaching ________.12.As the -values approach from the left, the slope is approaching _________.

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Analytically

13.On page 1.5, enter limh 0

f1(2 + h) f1(2)

h. Record

f (2) = lim

h 0

f(2 + h) f(2)

h=____.

14.In the space below, show the algebra and calculus to justify your answer.a. Write the definition of the derivative.

b. Substitute for andf(2) .

c. Expand and simplify.

d. Factor.

e. Reduce.

f. Take the limit.

15.Enter limh 0

f1(x + h) f1(x)

h. Record f (x) = lim

h 0

f(x + h) f(x)

h=_____________.

In the space below, show the algebra and calculus to justify your answer.

a. Write the definition of the derivative.

b. Substitute forf(x + h) and f(x) .

c. Expand and simplify.

d. Factor.

e. Reduce.

f. Take the limit.

16.Does this make sense when x = 2? Why?____________________________________________________________________

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Repeat the steps above for each problem below. Press / to move to the next problem.

17.Find f (2) if f(x) =x2 +1. Use the Graphs & Geometry page and the firstCalculator page to graphically and numerically verify your answer.

Record f (2) = limh 0

f(2 + h) f(2)

h=____________.

18.In the space below, show the algebra and calculus to justify your answer.

19.Enter limh 0

f1(x + h) f1(x)


h 0

f(x + h) f(x)

h=_____________.


20.Does this make sense when ? Why?_____________________________________________________________________

Problem 2

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21.Find f (1) if f(x) =x2 +x . Use the Graphs & Geometry page and the firstCalculator page to graphically and numerically verify your answer.

Record f (1) = limh 0

f(1+ h) f(1)

h =____________.

22.In the space below, show the algebra and calculus to justify your answer.

23.Enter limh 0

f1(x + h) f1(x)


h 0

f(x + h) f(x)

h=_____________.


24.Does this make sense when x = 1? Why?___________________________________________________________________

Problem 3

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Summary

25. Numerically, _____________.

26.Analytically, _____________when f(x) =_________________.

27.Graphically, represents ________________________________________________________________________________________________________.

28.Analytically, limh 0

(x + h)4 x4

h=__________ =__________ when f(x) =________.

-------------------------------------------------------------------------------------------------------

29.Numerically, limh 0

(2 + h)3+ (2 + h)

2 (2)

3+ (2)

2( )h

=_____________.

30.Analytically, limh

0

(2 + h)3+ (2 + h)

2 (2)

3+ (2)

2( )

h

=______________________

when f(x) =_________________.

31.Graphically, limh 0

(2 + h)3+ (2 + h)

2 (2)

3+ (2)

2( )h

represents _______________

____________________________________________________________________.

32.Analytically, limh 0

(x + h)3+ (x + h)

2 (x)

3+ (x)

2( )h

=

_____________ =____________

when f(x) =_________________.

-------------------------------------------------------------------------------------------------------

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Constructing Trapezoidal Sums Lab 13

1. Open the document Lab 13 Trapezoidal Sums. Press / to move to page 1.2.2. Estimate the area between the -axis and f(x) = 9 x2 between x = 2 and x = 2

using four trapezoids. Press / to move to page 1.3.

Graphically

3. Page 1.3 shows the graph of f(x) = 9 x2 . Changethe defined values on page 1.3 to reflect the

problem. Since the given interval is x = 2 to

x = 2 , let a = 2 and b = 2 .

4. If the height of each trapezoid is the same, find the-value at the start and end of each height for eachtrapezoid. Enter these values as indicated.

5. The length of each base should be determined bythe -value at the respective -value. Is this true

for each trapezoid drawn? If not, re-enter the -

values until they are correct.

6. Sketch the trapezoids on the graph provided andrecord the value of the area approximation.

Numerically

7. In the space below, show how to compute the area of each trapezoid. Find the sum ofthe area of the trapezoids.

8. Is the sum computed above the same as the sum computed on page 1.3? If not, checkyour work with a classmate or the teacher. Revise to find the correct areas and sum.

Trap approx=_____________

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Analytically

9. In the space below, use function notation to compute the area of each trapezoid. Findthe sum of the area of the trapezoids using page 1.4, a calculator page.

10.Using function notation, write a single expression to compute the area approximationusing four trapezoids. Algebraically simplify this expression by factoring andcombining like terms.

11.Confirm that the expression above yields the same sum as computed on page 1.3. Ifnot, check your work with a classmate or the teacher. Revise to find the correct

expression and sum.

12.Describe the pattern that can be used to find an area approximation using trapezoids.

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13.Estimate the area between the -axis andf(x) = 4

x

2

between andusing four trapezoids. Graph the four trapezoids

and compute the value of the trapezoidapproximation using function notation.

14.Estimate the area between the -axis andbetween x = 1 and x = 2

using four trapezoids. Graph the four trapezoids


Problem 2

Problem 3



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15.Estimate the area between the -axis andf(x) =x2 + 2 between x = 2 and x = 3

using four trapezoids. Graph the four trapezoids


Summary

16.Draw four trapezoids that would approximate thearea between the scatter plot and the -axis.Then compute the value of the trapezoid

approximation using function notation.

17.Write an expression that will approximate thearea under the curve f(x) and the -axis using

the four trapezoids that have been drawn.


Problem 4

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Date_____________ Name___________________________________________


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Constructing Slopefields Lab 15

1. Open the document Lab 15 Constructing Slopefields. Press / to move to page 1.2.2. Find the particular solution to the differential equation dy

dx= 2x 1 given that the

point 1, 1( ) lies on the solution. Press / to move to page 1.3.

Graphically and Numerically

3. On page 1.3, use the minimized slider bar to numerically adjust the value ofxv to seethe slopefield for the differential equation

dy

dx= 2x 1 drawn at the given x-value.

Record the slope at each x-value in the column.

-5 1

-4 2

-3 3

-2 4

-1 5

0

4. Describe how the graph of the differential equation dydx

= 2x 1 is drawn.

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

5. Graphically, the parent function of the solution to the given differential equationappears to be

_____________________________________________________________________

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Analytically

6. In the space below, show the algebra and calculus to find the general solution to theseparable differential equation

dy

dx= 2x 1.

7. Find the particular solution to the differential equation dydx

= 2x 1 given when x = 1,

.

On page 1.3, adjust the corresponding slider bars

forxo and yo to plot the initial condition. Doubleclick on f1(x) and enter the particular solution

found above.

8.

Does the graph of the solution following theshape of the slope field and pass through 1, 1( ) ?If yes, sketch the solution on the provided graph.

___________________________________

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9. Find the particular solution to the differential equation dydx

=

1

x2given that the point

lies on the solution.

-5

-4

-3

-2

-1

0

1

2

3

4

5

10.Show the algebra and calculus to find the particular solution to the separabledifferential equation.

Problem 2

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11.Find the particular solution to the differential equation dydx

= cos(x)given that the

point lies on the solution.

-5

-4

-3

-2

-1

0

1

2

3

4

5

12.Show the algebra and calculus to find the particular solution to the separabledifferential equation.

Problem 3

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13.Find the particular solution to the differential equation dydx

= 2sin(2x)given that the

point lies on the solution.

-5

-4

-3

-2

-1

0

1

2

3

4

5

14. Show the algebra and calculus to find the particular solution to the separabledifferential equation.

Problem 4

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Summary

15.Sketch the slope field for the differential equation using the provided tableand graph. Find and graph the particular solution given that the point 1, 0( ) lies on

the solution.

-3

-2

-1

0

1

2

3

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Exploring Accumulation Functions Lab 16

1. Open the document Lab 16 Accumulation Functions. Press / to move to page 1.2.2. Investigate what function is created as the signed area under the function

f(x) = 4x

2

, beginning at the lower bound a = 0 , is accumulated. Press / tomove to page 1.3.

Graphically

3. On page 1.3 is a graph of f(x) = 4 x2 and the shaded region from a = 0 to x = bwhich represents the area between the curve and the x -axis. There is also a point

b,I( ) whose x -coordinate is the value of b and whose y -coordinate is the definiteintegral of f(x) from x = a to x = b . Grab and drag the open circle (point b) on

the x -axis and observe the location of the plotted point b,I( ) .

Usegreater than, less than and equal to for the first blank; below, above, and neitherfor the second blank; andpositive, negative, andzero for the third blank.

4. Between the origin and the positive x -intercept, the value of b is _______________the value of a , the area between the graph of f(x) and the x -axis is

____________________ the x -axis, and the value of the definite integral I( ) is

____________________.

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5. Between the origin and the negative x -intercept, the value of b is_______________the value of a , the area between the graph of f(x) and

the x -axis is ____________________ thex -axis, and the value of the definite

integral (I) is ____________________.

6. When b is at the origin, the value of b is _______________the value of a , the areabetween the graph of f(x) and the x -axis is ____________________ thex -axis,

and the value of the definite integral (I) is ____________________.

7. How are the area and the definite integral related? Does it matter whether the valueofb is greater than, less than, or equal to the value of a ? Explain your answer.

_____________________________________________________________________

_____________________________________________________________________

8. Between the positive x -intercept and positive infinity, what happens to the area andthe value of the definite integral?

____________________________________________________________________

9. Between the negative x -intercept and negative infinity, what happens to the area andthe value of the definite integral?

_____________________________________________________________________

10.Explain how the area being above or below the x -axis and whetherb is greater than,less than, or equal to the value of a affects the value of the definite integral.

_____________________________________________________________________

_____________________________________________________________________

11.How does this explain the statement that the definite integral is the signed area?_____________________________________________________________________

12.Would there be values ofb other than when a = b for which the definite integral iszero? How would you estimate the values ofb at which the definite integral is zero?

____________________________________________________________________

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Graphically, Numerically and Analytically

13.For any value of b , there is a value for the definite integral, which creates thefunction g(x) = f(t) dt

0

x

. The upper bound x will be the different values of b and

g(x) will be the values ofI, the value of the definite integral. Press / ^ tocollect and plot the current point b,I( ) which is x,g(x)( ) .

14.On page 1.4, the collected values are shown in a List &Spreadsheet page.Plot enough ordered pairs on the grid to

sketch a graph ofg(x) = f(t) dt0

x

.

15.Press / G and enter a guess for theplotted function by editing f2(x) . This

function represents g(x) = f(t) dt=0

x

______________________________.

On page 1.5 enter 4 t2( ) dt0

x

to confirmyour guess.

16.Move to page 1.6, which is a graph ofg(x) = f(t) dt0

x

entered in f2(x) . Grab anddrag the point x and observe the slope of the curve. Press

/ ^to collect and plot

the current point x,m( ) .17.On page 1.7, the collected values are shown in a List & Spreadsheet page.

Plot enough ordered pairs on the grid to

sketch a graph of g(x) =d

dxf(t) dt

0

x

.

18.Press / G and enter a guess for the plottedfunction in f3(x) . This function

represents g(x) =d

dxf(t) dt

0

x

=

_______________________________.

On page 1.8 enterd

dx4 t2( ) dt

0

x

to

confirm your guess.

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For the following exercises, as the value of b is dragged, the points on the accumulationfunction are automatically plotted. Sketch the accumulation function, guess g(x) then verify

that g(x) fits the data, and calculate g'(x) using a calculator page. Press / to move tothe next problem.

19.Investigate what function is created as the signed area underthe function f(x) = 4 x2 , beginning at the

lower bound a = 1 , is accumulated.

g(x) = f(t) dt=1

x

___________________

g(x)=

d

dx f(t) dt=

1

x

________________

20.Investigate what function is created as the signed area underthe function f(x) = 4 x2 , beginning at the

lower bound a = 2 , is accumulated.

g(x) = f(t) dt=2

x

__________________

g(x) =d

dxf(t) dt=

2

x

_______________

21.Describe how the value of the lower bound a creates any differences or similaritiesbetween the accumulation functions and their rates of change.

_____________________________________________________________________

_____________________________________________________________________

Problem 3

Problem 2

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22.Investigate what function is created as the signed area under the functionf(x) = sin(2x) , beginning at the lower bound a =

2, is accumulated.

g(x) = f(t) dt=

2

x

____________________

g(x) =d

dxf(t) dt=

2

x

________________

23.Investigate what function is created as the signed area under the functionf(x) = e

x , beginning at the lower bound a = 0.5 , is accumulated.

g(x) = f(t) dt=0.5

x

___________________

g(x) =d

dx

f(t) dt=0.5

x

_______________

Problem 5

Problem 4

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Summary

24.Explain the relationship between f(x) and g(x) ._____________________________________________________________________

_____________________________________________________________________

25.Explain the relationship between f(x) and g(x) ._____________________________________________________________________

_____________________________________________________________________

26.How does the lower bound affect the graph of the accumulation function?_____________________________________________________________________

_____________________________________________________________________

27.How does the lower bound affect the graph of the derivative of the accumulationfunction?

_____________________________________________________________________

_____________________________________________________________________

28.Sketch a graph of the accumulation functiong(x) = f(t) dt

0

x

using the graph shown.Describe how you created the graph ofg(x) .

___________________________________

___________________________________

___________________________________

___________________________________

29.How would the graphs of f(t) dt0

x

and f(t) dt1x

compare?

_____________________________________________________________________

_____________________________________________________________________

WSNH_T3_2012

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