Resources

Resources

Intermediate Value Theorem

Which sign makes the statement true?

\newline

\frac{2}{5}

\text{?}

\frac{4}{5}

\newline

\newline

>

\newline

<

\newline

=

\newline

(-32)^{\frac{-4}{5}}

\newline

k(x)=\left\{\begin{array}{c}2 x^{2}-4 \text { for } x<2 \\ -3 x+10 \text { for } x \geq 2\end{array}\right.

\newline

k(x)

x=2

\newline

j(x)=k(x+1)

, what is the function for

j(x)

f(x)

5

over every unit interval in

x

f(0) = 0

\newline

Which could be a function rule for

f(x)

\newline

\newline

f(x) = -\frac{x}{5}

\newline

f(x) = 5x

\newline

f(x) = 5^x

\newline

f(x)

x

f(t)

4

over every unit interval in

t

f(0) = 0

\newline

Which could be a function rule for

f(t)

\newline

\newline

f(t) = 4t

\newline

f(t) = -t - 4

\newline

f(t) = -t + 4

\newline

f(t) = -\frac{t}{4}

f(t)

increases by a factor of

10

over every unit interval in

t

f(0) = 1

. Write a function rule for

f(t)

f(x)

6

over every unit interval in

x

f(0) = 0

\newline

Which could be a function rule for

f(x)

\newline

\newline

f(x) = x - 6

\newline

f(x) = x + 6

\newline

f(x) = 6x

\newline

f(x) = -6x

g(x)

2

over every unit interval in

x

g(0) = 0

\newline

Which could be a function rule for

g(x)

\newline

\newline

g(x) = -x + 2

\newline

g(x) = -2x

\newline

g(x) = 2x

\newline

g(x) = -\frac{x}{2}

g(x)

increases by a factor of

2

over every unit interval in

x

g(0) = 1

\newline

Write a function rule for

g(x)

g(t)

increases by a factor of

7

over every unit interval in

t

g(0) = 1

\newline

Which could be a function rule for

g(t)

\newline

\newline

g(t) = -7t

\newline

g(t) = 7^t

\newline

g(t) = 1 - 7t

\newline

g(t) = 1.07^t

g(x)

increases by a factor of

7

over every unit interval in

x

g(0) = 1

\newline

Which could be a function rule for

g(x)

\newline

\newline

g(x) = 7^x

\newline

g(x) = 1.07^x

\newline

g(x) = 1 + \frac{1}{7^x}

\newline

g(x)

1

7x

h(x)

1

over every unit interval in

x

h(0) = 0

\newline

Which could be a function rule for

h(x)

\newline

\newline

h(x) = x - 1

\newline

h(x) = x

\newline

h(x) = -x

\newline

h(x) = -x + 1

Determine if the following is a function, and if it is a one-to-one function.

\newline

\{(5,9),(9,2),(8,3),(0,8),(0,4)\}

\newline

\newline

A function, not one-to-one

\newline

A one-to-one function

Determine if the following is a function, and if it is a one-to-one function.

\newline

\{(8,9),(4,7),(3,8),(5,5),(2,0)\}

\newline

\newline

A function, not one-to-one

\newline

A one-to-one function

Determine if the following is a function, and if it is a one-to-one function.

\newline

\{(9,0),(7,7),(7,5),(5,9),(8,8)\}

\newline

\newline

A function, not one-to-one

\newline

A one-to-one function

Determine if the following is a function, and if it is a one-to-one function.

\newline

\{(9,3),(8,7),(3,2),(8,8),(2,5)\}

\newline

\newline

A function, not one-to-one

\newline

A one-to-one function

0<x<1

y=\log _{7} \frac{1}{x}

\frac{1}{y}=

Determine the average value of the piecewise function

f

defined below on the interval

x=-2

x=1

. Express your answer in simplest form.

\newline

f(x)=\left\{\begin{array}{lll} 3 x^{2}-8 & \text { for } & x \leq-1 \\ -3 x^{2} & \text { for } & x>-1 \end{array}\right.

g(x) = \frac{x-5}{\sqrt{x-4}-1}

x \neq 5

g

is continuous for all

x > 4

\newline

g(5)

Which of the following functions are continuous for all real numbers?

\newline

g(x)=\ln(x)

\newline

f(x)=\frac{1}{x}

\newline

1

\newline

g

\newline

f

\newline

g

f

\newline

g

f

Which of the following functions are continuous for all real numbers?

\newline

g(x)=\ln(x)

\newline

f(x)=\frac{1}{x}

\newline

1

\newline

g

\newline

f

\newline

g

f

\newline

g

f

f

be a continuous function on the closed interval

[-2,1]

f(-2)=3

f(1)=6

. Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=0

for at least one

c

-2

1

\newline

f(c)=4

for at least one

c

-2

1

\newline

f(c)=0

for at least one

c

[-2,1]

4

[-2,1]

5

\newline

f(c)=4

for at least one

c

[-2,1]

4

[-2,1]

5

Determine whether the function

f(x)

is continuous at

x=3

\newline

f(x)=\left\{\begin{array}{ll} 19-4 x^{2}, & x \geq 3 \\ -8-3 x, & x<3 \end{array}\right.

\newline

f(x)

is discontinuous at

x=3

\newline

f(x)

is continuous at

x=3

Determine whether the function

f(x)

is continuous at

x=-2

\newline

f(x)=\left\{\begin{array}{ll} 12-3 x^{2}, & x \leq-2 \\ -5-x, & x>-2 \end{array}\right.

\newline

f(x)

is continuous at

x=-2

\newline

f(x)

is discontinuous at

x=-2

Determine whether the function

f(x)

is continuous at

x=4

\newline

f(x)=\left\{\begin{array}{ll} 11-2 x^{2}, & x \leq 4 \\ -16-x, & x>4 \end{array}\right.

\newline

f(x)

is discontinuous at

x=4

\newline

f(x)

is continuous at

x=4

Determine whether the function

f(x)

is continuous at

x=-3

\newline

f(x)=\left\{\begin{array}{ll} 4+2 x^{2}, & x \leq-3 \\ 13-3 x, & x>-3 \end{array}\right.

\newline

f(x)

is continuous at

x=-3

\newline

f(x)

is discontinuous at

x=-3

Which of the following functions are continuous for all real numbers?

\newline

g(x)=\ln(x)

\newline

f(x)=\frac{1}{x}

\newline

1

\newline

g

\newline

f

\newline

g

f

\newline

g

f

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=2 y-x y

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=e^{x+y}

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=\frac{\cos (x)}{3 y-y^{2}}

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=3 e^{x y}

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=2 \sin (x)-3 \cos (y)

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=\sqrt{7 x+y^{2}}

\newline

1

\newline

\newline

Can this differential equation be solved using separation of variables?

\newline

\frac{d y}{d x}=\sin (3 x-y)

\newline

1

\newline

\newline

What is the area of the region between the graphs of

f(x)=x^{2}+2 x

g(x)=2 x+1

\newline

1

\newline

\frac{2}{3}

\newline

2

\newline

\frac{4}{3}

\newline

\frac{10}{3}

f

be a continuous function on the closed interval

[-2,1]

f(-2)=3

f(1)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=4

for at least one

c

-2

1

\newline

f(c)=4

for at least one

c

3

6

\newline

f(c)=0

for at least one

c

-2

1

\newline

f(c)=0

for at least one

c

3

6

f

be a continuous function on the closed interval

[-5,0]

f(-5)=0

f(0)=5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=2

for at least one

c

-5

0

\newline

f(c)=-2

for at least one

c

0

5

\newline

f(c)=2

for at least one

c

0

5

\newline

f(c)=-2

for at least one

c

-5

0

g

be a continuous function on the closed interval

[-1,3]

g(-1)=-2

g(3)=-5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-3

for at least one

c

-1

3

\newline

g(c)=0

for at least one

c

-5

-2

\newline

g(c)=-3

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-1

3

g

be a continuous function on the closed interval

[-1,4]

g(-1)=-4

g(4)=1

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=3

for at least one

c

-4

1

\newline

g(c)=3

for at least one

c

-1

4

\newline

g(c)=-3

for at least one

c

-1

4

\newline

g(c)=-3

for at least one

c

-4

1

g

be a continuous function on the closed interval

[-3,3]

g(-3)=0

g(3)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-2

for at least one

c

0

6

\newline

g(c)=5

for at least one

c

0

6

\newline

g(c)=5

for at least one

c

-3

3

\newline

g(c)=-2

for at least one

c

-3

3

h

be a continuous function on the closed interval

[-3,4]

h(-3)=-1

h(4)=2

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

h(c)=-2

for at least one

c

-1

2

\newline

h(c)=1

for at least one

c

-3

4

\newline

h(c)=1

for at least one

c

-1

2

\newline

h(c)=-2

for at least one

c

-3

4

h

be a continuous function on the closed interval

[0,4]

h(0)=2

h(4)=-2

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

h(c)=3

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

0

4

\newline

h(c)=-1

for at least one

c

-2

2

\newline

h(c)=3

for at least one

c

-2

2

f(x)=\sin (x)

\newline

Can we use the intermediate value theorem to say the equation

f(x)=0

has a solution where

\frac{\pi}{6} \leq x \leq \frac{\pi}{3}

\newline

1

\newline

(A) No, since the function is not continuous on that interval.

\newline

0

f\left(\frac{\pi}{6}\right)

f\left(\frac{\pi}{3}\right)

\newline

(C) Yes, both conditions for using the intermediate value theorem have been met.

h(x)=\frac{1}{x}

\newline

Can we use the intermediate value theorem to say the equation

h(x)=0.5

has a solution where

-1 \leq x \leq 1

\newline

1

\newline

(A) No, since the function is not continuous on that interval.

\newline

0

5

h(-1)

h(1)

\newline

(C) Yes, both conditions for using the intermediate value theorem have been met.

f

be a continuous function on the closed interval

[1,5]

f(1)=1

f(5)=-3

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=2

for at least one

c

-3

1

\newline

f(c)=-2

for at least one

c

-3

1

\newline

f(c)=2

for at least one

c

1

5

\newline

f(c)=-2

for at least one

c

1

5

g

be a continuous function on the closed interval

[-3,3]

g(-3)=0

g(3)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-2

for at least one

c

-3

3

\newline

g(c)=-2

for at least one

c

0

6

\newline

g(c)=5

for at least one

c

-3

3

\newline

g(c)=5

for at least one

c

0

6

h

be a continuous function on the closed interval

[-3,4]

h(-3)=-1

h(4)=2

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

h(c)=1

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-3

4

\newline

h(c)=-2

for at least one

c

-1

2

\newline

h(c)=1

for at least one

c

-1

2

f

be a continuous function on the closed interval

[-5,0]

f(-5)=0

f(0)=5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=-2

for at least one

c

-5

0

\newline

f(c)=2

for at least one

c

0

5

\newline

f(c)=2

for at least one

c

-5

0

\newline

f(c)=-2

for at least one

c

0

5

f

be a continuous function on the closed interval

[-2,1]

f(-2)=3

f(1)=6

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

f(c)=0

for at least one

c

-2

1

\newline

f(c)=0

for at least one

c

3

6

\newline

f(c)=4

for at least one

c

-2

1

\newline

f(c)=4

for at least one

c

3

6

g

be a continuous function on the closed interval

[-1,3]

g(-1)=-2

g(3)=-5

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-3

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-5

-2

\newline

g(c)=0

for at least one

c

-1

3

\newline

g(c)=-3

for at least one

c

-1

3