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Lesson 1 Question 1

For the following function, , evaluate the following limits:

Lesson 1 Question 2

Consider the piecewise function

- a) Graph the function on the domain [-3, 3].

- b) Find the .

- c) Find the .

- d) Find the .

- e) Find .

Lesson 1 Question 3

Sketch a possible graph of a function that has domain , has zeros (intersects the x-axis) at -2, and 1, and the , and

Possible answer:

Lesson 1 Question 4

Sketch a possible graph of a function that has domain , but not , goes through the points (-4,2) and (0,2), and has .

Possible answer:

Lesson 1 Question 5

Sketch a possible graph of a function that has zeros of -2, and 1, and , but , and

Possible answer:

Lesson 1 Question 6

Estimate the value of by trying values very close to 0 (from both the right and the left, obviously, since it's not a one-sided limit). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Note: The limit value converges to "e"

Lesson 1 Question 7

Estimate the value of by trying values very close to 2 (from the left only, of course). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Estimation:

Graph:

Lesson 1 Question 8

For the function , approximate the slope of the tangent line at (1,3) by doing the following:

- Make a graph of f(x) and plot a point at , somewhere nearby the point (1,3).

- Write the slope of the secant line that joins the point (1,3) and the point (a, f(a)) in terms of a, and simplify.

- Take the to determine the actual slope of the tangent line at (1,3).

Lesson 1 Question 9

For the function , approximate the slope of the tangent line at (1,- 2) using the secant line approximation method (see question ID 158 for the individual steps).

Lesson 1 Question 10

Evaluate the following limit:

Lesson 1 Question 11

Evaluate this limit:

Lesson 1 Question 12

Evaluate the following limit:

Lesson 1 Question 13

Evaluate this limit:

Lesson 1 Question 14

Evaluate this limit:

Lesson 1 Question 15

Evaluate the following limit:

Lesson 1 Question 16

AP Prep:

Which of the following limits exist?

Lesson 1 Question 17

AP Prep:

Consider the function:

Which of the following are true?

- exists.

- exists.

Lesson 1 Question 18

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that .

Lesson 1 Question 19

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that .

Lesson 1 Question 20

[Try This!]:

is equivalent to:

(a) For all , there is an such that if , then .

(b) For all , there is a such that if , then .

(c) For all , there is a such that if , then .

(d) For all , there is an such that if , then .

Lesson 2 Question 1

Estimate the area under the following curve using the suggested Rectangular Approximation Method.

on the interval [0,2] (Use the a) Left End Point and b) Right End Point for the height in each interval, and 4 rectangles)

- Left End Point

- Right End Point

Lesson 2 Question 2

Estimate the area under the following curve using the suggested Rectangular Approximation Method.

on the interval [0, ] (Use the Mid Point for the height in each interval, and 3 rectangles)

Lesson 2 Question 3

Estimate the area under the following curve using the suggested Rectangular Approximation Method.

on the interval [-3, -1] (Use the (a) Left, (b) Right, and (c) Middle endpoint for the height in each interval, and 5 rectangles)

- Left

- Right

- Middle endpoint

Lesson 2 Question 4

Evaluate the following to practice the basic skills of Sigma Notation.

Lesson 2 Question 5

Evaluate the following to practice the basic skills of Sigma Notation.

Lesson 2 Question 6

Write in sigma notation (but don't evaluate):

Lesson 2 Question 7

Divide on the interval into 4 subintervals of equal size and then compute with as the left point of each subinterval

Lesson 2 Question 8

Divide on the interval into 4 subintervals of equal size and then compute with as the right endpoint of each subinterval.

Lesson 2 Question 9

Divide on the interval into 4 subintervals of equal size and then compute with as the midpoint of each subinterval.

Lesson 2 Question 10

If you have a special program in your graphing calculator to do this (yes, you can find those! You can also program it yourself using the PRGM function... ), find the area under the curve on [1,2] using left end, right end, and midpoint, approximations for 10, 20 and 50 divisions.

Lesson 2 Question 11

Find

Lesson 2 Question 12

Find

Lesson 2 Question 13

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between and . Which sum gives an overestimate?

Lesson 2 Question 14

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between and . Which sum gives an underestimate?

Lesson 2 Question 15

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between and . Which sum gives an overestimate?

Lesson 2 Question 16

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between and . Which sum gives an underestimate?

Lesson 3 Question 1

Find the volume when the area bounded by the line and the curve are rotated about the x axis.

Lesson 3 Question 2

Find the volume when the area bounded by the lines , , and are rotated about the x--axis.

Lesson 3 Question 3

Find the volume of revolution when the area bounded by the curve , is rotated about the y-axis.

Lesson 3 Question 4

Find the volume of a sphere with a radius of 3 units using the volume by discs method.

Lesson 3 Question 5

Lesson 3 Question 6

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

from to

Lesson 3 Question 7

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

Lesson 3 Question 8

Lesson 3 Question 9

For the following question, use technology to find the approximate Volume of the solid that results when the indicated areas are rotated about the y axis.

, ,

Lesson 3 Question 10

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the line .

Lesson 3 Question 11

Lesson 3 Question 12

For the following questions, find the Volume of the solid that results when the indicated areas are rotated about the y-axis.

Lesson 3 Question 13

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the y-axis.

Lesson 3 Question 14

Lesson 3 Question 15

For the following question, find the Volume of the solid that results when the indicated area is rotated about the y-axis.

Lesson 3 Question 16

Find the volume when a solid bounded by the curves , , and is revolved around the line .

Lesson 3 Question 17

Find the volume when a solid bounded by the curves and and are revolved around the line .

Lesson 3 Question 18

Find the volume of a solid created by rotating the area bounded by and are revolved around the y-axis.

Lesson 3 Question 19

The region bounded between the curves and is rotated about the line . Using the disc/washer method (i.e. not cylindrical shells), determine the integral used to calculate the volume of the resulting solid. (Note: Set up the integral, but you do not have to evaluate.)

Lesson 3 Question 20

Lesson 3 Question 21

AP Prep:

What is the volume of the solid generated by rotating the area enclosed by , , the x-axis, and , around the x-axis.

Lesson 3 Question 22

AP Prep:

Find the volume of the solid generated by rotating the region bounded by the x-axis, y-axis, , , and around the x-axis.

Lesson 3 Question 23

AP Prep:

Find the volume of a damaged cardboard box with a base we can model as the region under the parabola bounded by the y-axis, the x-axis, and , and cross sections perpendicular to the x-axis which are squares.

Lesson 3 Question 24

AP Prep:

Find the volume of the solid generated by rotating the ellipse about the y-axis.

Lesson 3 Question 25

AP Prep:

The base of a solid is the region enclosed by the parabola , the line , and the x-axis. If the cross sections perpendicular to the x-axis are squares, find its volume.

Lesson 3 Question 26

AP Prep:

The base of a solid is the region in the first quadrant enclosed by the graph , and the line . If the cross sections perpendicular to the x-axis are semi-circles, find its volume.

Lesson 3 Question 27

AP Prep:

The base of a solid is the region enclosed by the graph of , the line , and the y-axis. If the cross sections perpendicular to the y-axis are equilateral triangles, find its volume.

Lesson 3 Question 28

AP Prep:

The base of a solid is the region in the first quadrant bounded by the x-axis, y-axis and the line .If the cross sections perpendicular to the y-axis are rectangles whose width is half their length (where the length is in the x direction), find its volume.

Lesson 4 Question 1

The figure below shows the graphs of f(x), f'(x), f''(x). Identify each curve, and explain your choices.

Lesson 4 Question 2

Lesson 4 Question 3

Lesson 4 Question 4

Lesson 4 Question 5

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