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Lesson 1 Question 1

For the following function, f(x), evaluate the following limits: 

  1. \displaystyle\mathop{{\rm lim}}_{x\to {-1}^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to {-1}^+}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to -1}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 0^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 0^+}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 0}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 1^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 1^+}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 1}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 2^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 2^+}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 2}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 3^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 3^+}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 3}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 5^-}f\left(x\right)

  1. \displaystyle\mathop{{\rm lim}}_{x\to 5^+}f\left(x\right)

Studyforge Question ID: 19085

Lesson 1 Question 2

Consider the piecewise function
f(x)=\begin{cases}\dfrac{x^4-1}{x-1} & x\neq 1\\3 & x=1\\\end{cases}

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

  1. b) Find the \displaystyle\mathop{{\rm lim}}_{x\to 1^-}f\left(x\right) .

  1. c) Find the \displaystyle\mathop{{\rm lim}}_{x\to 1^+}f\left(x\right) .

  1. d) Find the \displaystyle\mathop{{\rm lim}}_{x\to 1}f\left(x\right).

  1. e) Find f(1).

Studyforge Question ID: 151

Lesson 1 Question 3

Sketch a possible graph of a function that has domain -3\le x\le 5, has zeros (intersects the x-axis) at -2, and 1, and the \displaystyle\lim_{x\to -2}f(x)=5, and \displaystyle\lim_{x\to 1}f(x)=DNE

Possible answer:

Studyforge Question ID: 153

Lesson 1 Question 4

Sketch a possible graph of a function that has domain  x \le 0, but not x=-2,  goes through the points (-4,2) and (0,2), and has \displaystyle\mathop{{\rm lim}}_{x\to -2}f\left(x\right)=\ \infty.

Possible answer:

Studyforge Question ID: 154

Lesson 1 Question 5

Sketch a possible graph of a function that has zeros of -2, and 1, and \displaystyle\lim_{x\to -2^-}f(x)=-\infty , but \displaystyle\lim_{x\to -2^+}f(x)=\infty, and \displaystyle\lim_{x\to 1}f(x)=3

Possible answer:

Studyforge Question ID: 155

Lesson 1 Question 6

Estimate the value of \mathop{{\rm lim}}_{x\to 0}(1+x)^{\frac{1}{x}} 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"

Studyforge Question ID: 156

Lesson 1 Question 7

Estimate the value of \mathop{{\rm lim}}_{x\to 2^-}\frac{3x+6}{3x^3-24} 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: 

Studyforge Question ID: 157

Lesson 1 Question 8

For the function f(x)={3x}^2, approximate the slope of the tangent line at (1,3) by doing the following:

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

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

  1. Take the \mathop{{\rm lim}}_{a\to 1}\ m_{sec} to determine the actual slope of the tangent line at (1,3).

Studyforge Question ID: 158

Lesson 1 Question 9

For the function f\left(x\right)=-2x^2 , approximate the slope of the tangent line at (1,- 2) using the secant line approximation method (see question ID 158 for the individual steps).

Studyforge Question ID: 159

Lesson 1 Question 10

Evaluate the following limit:

\displaystyle\mathop{{\rm lim}}_{x\to 5}\ \ 3x

Studyforge Question ID: 160

Lesson 1 Question 11

Evaluate this limit:

\displaystyle\mathop{{\rm lim}}_{x\to \pi }\ \ 4x^2

Studyforge Question ID: 161

Lesson 1 Question 12

Evaluate the following limit:

\displaystyle\mathop{{\rm lim}}_{x\to 0}\frac{{\rm 8}}{x}

Studyforge Question ID: 162

Lesson 1 Question 13

Evaluate this limit:

\displaystyle\lim_{x\to 0}\frac{\cos{(x)}}{x}

Studyforge Question ID: 163

Lesson 1 Question 14

Evaluate this limit:

\displaystyle\lim_{x\to 0^+}\frac{5}{x^2}

Studyforge Question ID: 164

Lesson 1 Question 15

Evaluate the following limit:

\displaystyle\lim_{x\to 0^-}\frac{10}{x^3}

Studyforge Question ID: 165

Lesson 1 Question 16

AP Prep:

Which of the following limits exist?

  1. \lim_{x\to -5^{-}}{f(x)}

  1. \lim_{x\to -5^{+}}{f(x)}

  1. \lim_{x\to -5}{f(x)}

  1. \lim_{x\to -3^{-}}{f(x)}

  1. \lim_{x\to -3^{+}}{f(x)}

  1. \lim_{x\to -3}{f(x)}

  1. \lim_{x\to 0^{-}}{f(x)}

  1. \lim_{x\to 0^{+}}{f(x)}

  1. \lim_{x\to 0}{f(x)}

  1. \lim_{x\to 3^{-}}{f(x)}

  1. \lim_{x\to 3^{+}}{f(x)}

  1. \lim_{x\to 3}{f(x)}

  1. \lim_{x\to 5^{-}}{f(x)}

  1. \lim_{x\to 5^{+}}{f(x)}

  1. \lim_{x\to 5}{f(x)}

Studyforge Question ID: 3207

Lesson 1 Question 17

AP Prep:

Consider the function:

f(x)=\begin{cases}x^2+2 & x<3\\x & x\ge 3\end{cases}

Which of the following are true?

  1. f(3)=11

  1. \lim_{x\to 3}f(x)=3

  1. \lim_{x\to 3^{-}}f(x) exists.

  1. \lim_{x\to 3}f(x) exists.

  1. \lim_{x\to 3^{+}}f(x)=3

  1. \lim_{x\to 3^{+}}f(x)=f(3)

Studyforge Question ID: 9149

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 \lim_{x\to 2}{2x+2}=6.

Studyforge Question ID: 3208

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 \lim_{x\to 1}{3x-2}=1.

Studyforge Question ID: 9130

Lesson 1 Question 20

[Try This!]:

\lim_{x\to c}f(x)=L is equivalent to:

(a) For all \delta, there is an \epsilon such that if |x-c|<\delta, then |f(x)-L|<\epsilon.

(b) For all \epsilon, there is a \delta such that if |x-c|>\delta, then |f(x)-L|<\epsilon.

(c) For all \epsilon, there is a \delta such that if |x-c|<\delta, then |f(x)-L|<\epsilon.

(d) For all \delta, there is an \epsilon such that if |f(x)-L|<\epsilon, then |x-c|<\delta.

Studyforge Question ID: 9148

Lesson 2 Question 1

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

f\left(x\right)=\sqrt[3]{x} 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)

  1. Left End Point

  1. Right End Point

Studyforge Question ID: 378

Lesson 2 Question 2

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

f\left(x\right)=\cos{x} on the interval [0, \pi] (Use the Mid Point for the height in each interval, and 3 rectangles)

Studyforge Question ID: 379

Lesson 2 Question 3

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

f\left(x\right)=\frac{1}{2x^2} on the interval [-3, -1] (Use the (a) Left, (b) Right, and (c) Middle endpoint for the height in each interval, and 5 rectangles)

  1. Left

  1. Right

  1. Middle endpoint

Studyforge Question ID: 380

Lesson 2 Question 4

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

\displaystyle\sum^3_{i=1}{i^4}

Studyforge Question ID: 381

Lesson 2 Question 5

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

\displaystyle\sum^1_{k=-4}{\left(2k^2-2k+1\right)}

Studyforge Question ID: 382

Lesson 2 Question 6

Write in sigma notation (but don't evaluate): 3+6+9+12+...+21

Studyforge Question ID: 383

Lesson 2 Question 7

Divide f\left(x\right)=2x+2 on the interval \left[-1,3\right] into 4 subintervals of equal size and then compute \sum^4_{k=1}{f(x^*_k})\Delta x with x*_k as the left point of each subinterval

Studyforge Question ID: 384

Lesson 2 Question 8

Divide f\left(x\right)=2x+2 on the interval \left[-1,3\right] into 4 subintervals of equal size and then compute \sum^4_{k=1}{f(x^*_k})\Delta x with x*_k as the right endpoint of each subinterval.

Studyforge Question ID: 385

Lesson 2 Question 9

Divide f\left(x\right)=2x+2 on the interval \left[-1,3\right] into 4 subintervals of equal size and then compute \sum^4_{k=1}{f(x^*_k})\Delta x with x*_k as the midpoint of each subinterval.

Studyforge Question ID: 386

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 f\left(x\right)=\frac{1}{x^2} on [1,2] using left end, right end, and midpoint, approximations for 10, 20 and 50 divisions.

Studyforge Question ID: 387

Lesson 2 Question 11

Find \sum^b_{i=a}{f(x*_i)\triangle x}

f\left(x\right)=x-1;\ a=0;\ b=4;\ n=4;\ \triangle x_1=2;\ \triangle x_2=1;\ \triangle x_3=0.7;\ \triangle x_4=0.3;\ x*_1=1.5;\ x*_2=1.1;\ x^*_3=2.5;\ x^*_4=2.8

Studyforge Question ID: 388

Lesson 2 Question 12

Find \sum^b_{i=a}{f(x*_i)\triangle x}

f\left(x\right)=x^2-1;\ a=-3;\ b=5;\ n=3;\ \triangle x_1=2;\ \triangle x_2=1;\ \triangle x_3=5;\ x^*_1=-1.5;\ {\ x}^*_2=-0.1;\ x^*_3=3.5

Studyforge Question ID: 389

Lesson 2 Question 13

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between x=0 and x=1. Which sum gives an overestimate?

Studyforge Question ID: 9219

Lesson 2 Question 14

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between x=0 and x=1. Which sum gives an underestimate?

Studyforge Question ID: 9220

Lesson 2 Question 15

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between x=1 and x=2. Which sum gives an overestimate?

Studyforge Question ID: 9221

Lesson 2 Question 16

AP Prep:

A right and left Riemann sum are used to approximate the area under the function depicted below between x=1 and x=2. Which sum gives an underestimate?

Studyforge Question ID: 9222

Lesson 3 Question 1

Find the volume when the area bounded by the line y=2x and the curve y=6x-x^2 are rotated about the x axis.

Studyforge Question ID: 607

Lesson 3 Question 2

Find the volume when the area bounded by the lines y=2-3x, x=0, and y=0 are rotated about the x--axis.

Studyforge Question ID: 608

Lesson 3 Question 3

Find the volume of revolution when the area bounded by the curve y=2-3x, x=0,\text{ and }y=0 is rotated about the y-axis. 

Studyforge Question ID: 609

Lesson 3 Question 4

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

Studyforge Question ID: 610

Lesson 3 Question 5

(Try this!): Find the volume of a solid built on a base which is bounded by the curves y=\sqrt{x-1} and the x axis from 1 to 5, if the cross sections perpendicular to the x axis are squares.

Studyforge Question ID: 611

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.

y=x^{1/3}+1,\ y=1 from x=0 to x=8

Studyforge Question ID: 612

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.

y=4-x^2,\hspace{5mm}y=0

Studyforge Question ID: 613

Lesson 3 Question 8

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

y=\sqrt{\sin x},\ \ x=2y

Studyforge Question ID: 614

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.

x=\tan y, x=0, y=1

Studyforge Question ID: 615

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 y=-1.

y=2\cos x,\ y=-1,\ x=-1,\ x=1

Studyforge Question ID: 616

Lesson 3 Question 11

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

y=\frac{1}{\sqrt{x^2+9}}, y=0,\ \ x=0\ to\ 3

Studyforge Question ID: 617

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.

x=1/\sqrt{4+y}, x=0, y\ from\ 0\ to\ 5\

Studyforge Question ID: 618

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.

x={\sec y,\ } x=0 ,\ y\ from-\frac{\pi }{4}\ to\frac{\pi }{4}

Studyforge Question ID: 619

Lesson 3 Question 14

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

x=y^3,\ \ y=x

Studyforge Question ID: 620

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.

y=e^{3x},\ x=0,\ x=1,\ y=0

Studyforge Question ID: 621

Lesson 3 Question 16

Find the volume when a solid bounded by the curves y=3\sqrt{x-1}, y=0, and x=4 is revolved around the line y=-6.

Studyforge Question ID: 622

Lesson 3 Question 17

Find the volume when a solid bounded by the curves y=3\sqrt{x-1} and y=0 and x=5 are revolved around the line x=-4.

Studyforge Question ID: 623

Lesson 3 Question 18

Find the volume of a solid created by rotating the area bounded by x=2y-y^2, and x=y are revolved around the y-axis.

Studyforge Question ID: 624

Lesson 3 Question 19

The region bounded between the curves x=2y^2 and y=2x is rotated about the line x=2. 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.)

Studyforge Question ID: 625

Lesson 3 Question 20

(Try this!) Find the volume of the sphere created by rotating the top half of the circle defined by the equation x^2+y^2=9 over the x-axis.

Studyforge Question ID: 626

Lesson 3 Question 21

AP Prep:

What is the volume of the solid generated by rotating the area enclosed by f(x)=x^3, g(x)=2^{1-x}, the x-axis, and x=4, around the x-axis.

Studyforge Question ID: 9246

Lesson 3 Question 22

AP Prep:

Find the volume of the solid generated by rotating the region bounded by the x-axis, y-axis, x=2,  f(x)=e^{x-1}, and g(x)=e^{1-x} around the x-axis.

Studyforge Question ID: 9247

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 f(x)=\dfrac{(x-1)^2}{4}+1 bounded by the y-axis, the x-axis, and x=2, and cross sections perpendicular to the x-axis which are squares.

Studyforge Question ID: 9248

Lesson 3 Question 24

AP Prep:

Find the volume of the solid generated by rotating the ellipse 4x^2+y^2=9 about the y-axis.

Studyforge Question ID: 9249

Lesson 3 Question 25

AP Prep:

The base of a solid is the region enclosed by the parabola y = 3x^2, the line x = 2, and the x-axis. If the cross sections perpendicular to the x-axis are squares, find its volume.

Studyforge Question ID: 9604

Lesson 3 Question 26

AP Prep:

The base of a solid is the region in the first quadrant enclosed by the graph y = 2e^{-x}, and the line x = 1. If the cross sections perpendicular to the x-axis are semi-circles, find its volume.

Studyforge Question ID: 9605

Lesson 3 Question 27

AP Prep:

The base of a solid is the region enclosed by the graph of y = \sqrt{sin x}, the line y = 1, and the y-axis. If the cross sections perpendicular to the y-axis are equilateral triangles, find its volume.

Studyforge Question ID: 9606

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 x + 4y = 8.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.

Studyforge Question ID: 9607

Lesson 4 Question 1

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

Studyforge Question ID: 709

Lesson 4 Question 2

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

Studyforge Question ID: 710

Lesson 4 Question 3

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

Studyforge Question ID: 711

Lesson 4 Question 4

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

Studyforge Question ID: 712

Lesson 4 Question 5

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

Studyforge Question ID: 713
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