Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

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Example Questions

Example Question #71 : Calculus Ab

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Decide which of the following functions, most likely represents the sample data above.

Possible Answers:

\displaystyle \frac{(18x + 17)}{(12x - 11)}

\displaystyle -\frac{(18x + 17)}{(12x - 11)}

\displaystyle -\frac{(12x - 11)}{(18x + 17)}

\displaystyle \frac{(12x - 11)}{(18x + 17)}

Correct answer:

\displaystyle -\frac{(18x + 17)}{(12x - 11)}

Explanation:

Notice that as x increases, the points of data graphed appears to level out, flattening towards a certain value. This value is what is known as a horizontal asymptote. An asymptote is a value that a function approaches, but never actually reaches. Think of a horizontal asymptote as a limit of a function as x approaches infinity. In such a case, as x approaches infinity, any constants added or subtracted in the numerator and denominator become irrelevant. What matters is the power of x in the denominator and the numerator; if those are the same, then to coefficients define the asymptote. We see that the function flattens towards:

\displaystyle f\left ( \infty \right )=-1.5

This matches the ratio of coefficients for the function:

\displaystyle -\frac{(18x + 17)}{(12x - 11)}

Example Question #20 : Connect Infinite Limits And Vertical/Horizontal Asymptotes

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Which of the four following functions is most likely depicted in the sample data above?

Possible Answers:

\displaystyle -\frac{(x + 1)}{(10x + 13)}

\displaystyle -\frac{(x - 1)}{(10x - 13)}

\displaystyle -\frac{(10x - 13)}{(x - 1)}

\displaystyle -\frac{(10x + 13)}{(x + 1)}

Correct answer:

\displaystyle -\frac{(x + 1)}{(10x + 13)}

Explanation:

Observation of the data points shows that there is a sharp increase and decrease on either side of a particular x-value. This type of behavior is observed when there is a vertical asymptote. An asymptote is a value that a function may approach, but will never actually attain. In the case of vertical asymptotes, this behavior occurs if the function approaches infinity for a given x-value, often when a zero value appears in a denominator. Noting this rule, the above function has a zero in the denominator at a definite point centered approximately around:

\displaystyle x=-1.3

We find a zero denominator for the function:

\displaystyle -\frac{(x + 1)}{(10x + 13)}

Example Question #81 : Calculus Ab

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Which of the four following functions is probably depicted in the sample data above?

Possible Answers:

\displaystyle \frac{(19x - 15)}{(5x - 15)}

\displaystyle -\frac{(5x - 15)}{(19x - 15)}

\displaystyle \frac{(5x - 15)}{(19x - 15)}

\displaystyle -\frac{(19x - 15)}{(5x - 15)}

Correct answer:

\displaystyle -\frac{(19x - 15)}{(5x - 15)}

Explanation:

Notice that as x increases, the points of data graphed appears to level out, flattening towards a certain value. This value is what is known as a horizontal asymptote. An asymptote is a value that a function approaches, but never actually reaches. Think of a horizontal asymptote as a limit of a function as x approaches infinity. In such a case, as x approaches infinity, any constants added or subtracted in the numerator and denominator become irrelevant. What matters is the power of x in the denominator and the numerator; if those are the same, then to coefficients define the asymptote. We see that the function flattens towards: 

\displaystyle f\left ( \infty \right )=-3.8

This matches the ratio of coefficients for the function:

\displaystyle -\frac{(19x - 15)}{(5x - 15)}

Example Question #82 : Calculus Ab

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Decide which of the following functions, most likely represents the sample data above.

Possible Answers:

\displaystyle -\frac{(4x - 1)}{(x + 11)}

\displaystyle -\frac{(4x + 1)}{(x - 11)}

\displaystyle -\frac{(x + 11)}{(4x - 1)}

\displaystyle -\frac{(x - 11)}{(4x + 1)}

Correct answer:

\displaystyle -\frac{(x - 11)}{(4x + 1)}

Explanation:

Observation of the data points shows that there is a sharp increase and decrease on either side of a particular x-value. This type of behavior is observed when there is a vertical asymptote. An asymptote is a value that a function may approach, but will never actually attain. In the case of vertical asymptotes, this behavior occurs if the function approaches infinity for a given x-value, often when a zero value appears in a denominator. Noting this rule, the above function has a zero in the denominator at a definite point centered approximately around:

 \displaystyle x=-0.3

We find a zero denominator for the function:

\displaystyle -\frac{(x - 11)}{(4x + 1)}

Example Question #1 : Apply Intermediate Value Theorem

Which of the following does NOT satisfy the conditions required to apply the Intermediate Value Theorem to a function \displaystyle f on the interval \displaystyle [a,b]?

Possible Answers:

Intermediate Value Theorem considers points within the interval \displaystyle [a,b]

Intermediate Value Theorem cannot be applied outside of the interval \displaystyle [a,b]

 

Intermediate Value Theorem can be used when \displaystyle f(a) = f(b)

The function f must be continuous along \displaystyle [a,b] to apply Intermediate Value Theorem

Correct answer:

Intermediate Value Theorem can be used when \displaystyle f(a) = f(b)

Explanation:

Intermediate Value Theorem states that if the function \displaystyle f is continuous and has a domain containing the interval \displaystyle [a,b], then at some number \displaystyle c within the interval \displaystyle [a,b] the function will take on a value \displaystyle f(c)=N that is between the values of \displaystyle f(a) and \displaystyle f(b).

The conditions that must be satisfied in order to use Intermediate Value Theorem include that the function must be continuous and the number \displaystyle c must be within the interval \displaystyle [a,b]. However, \displaystyle f(a) cannot equal \displaystyle f(b).

Therefore, the answer choice “Intermediate Value Theorem can be used when \displaystyle f(a) = f(b)” does not satisfy the necessary conditions and is the correct answer for this question.

 

Example Question #1 : Apply Intermediate Value Theorem

Using Intermediate Value Theorem to analyze a continuous function, what can be deduced if a polynomial changes signs within an interval?

Possible Answers:

The function is not differentiable within that interval

There is a local maximum between the endpoints of that interval

The function is not continuous within that interval

 

There is a zero between the endpoints of that interval

Correct answer:

There is a zero between the endpoints of that interval

Explanation:

Intermediate Value Theorem is only true with continuous, differentiable functions, thus eliminating the answer choices “The function is not continuous within that interval” and “The function is not differentiable within that interval.” There is not necessarily a local maximum or minimum contained in the interval either. That leaves the correct answer choice, “There is a zero between the endpoints of that interval.” If the polynomial is changing signs and meets the requirements for Intermediate Value Theorem, it must cross the \displaystyle x axis at some point within the interval.

Example Question #1 : Apply Intermediate Value Theorem

What theorem could you use to show that a polynomial has a root on a given interval?

Possible Answers:

Extreme Value Theorem

Intermediate Value Theorem

 

Fundamental Theorem of Calculus

Mean Value Theorem for Derivatives

Correct answer:

Intermediate Value Theorem

 

Explanation:

A polynomial has a zero or root when it crosses the \displaystyle x axis. For a given interval \displaystyle [a,b], if a and b have different signs (for instance, if \displaystyle a is negative and \displaystyle b is positive), then by Intermediate Value Theorem there must be a value of zero between \displaystyle f(a) and \displaystyle f(b). Therefore, Intermediate Value Theorem is the correct answer.

Example Question #1 : Apply Intermediate Value Theorem

Using the continuous function \displaystyle f and the interval \displaystyle [a,b], which of the following correctly identifies why the Intermediate Value Theorem is useful?

Possible Answers:

The Intermediate Value Theorem tells you how many times the function \displaystyle f repeats a value as it progresses from \displaystyle f(a) to \displaystyle f(b)

The Intermediate Value Theorem states that somewhere between \displaystyle f(a) and \displaystyle f(b) there exists a value \displaystyle f(c)=N, with \displaystyle a< c< b

The Intermediate Value Theorem can identify the value of \displaystyle f(c)=N that the function takes on as it passes from \displaystyle f(a) to \displaystyle f(b)

The Intermediate Value Theorem is not useful

Correct answer:

The Intermediate Value Theorem states that somewhere between \displaystyle f(a) and \displaystyle f(b) there exists a value \displaystyle f(c)=N, with \displaystyle a< c< b

Explanation:

The Intermediate Value Theorem tells us that a value between \displaystyle f(a) and \displaystyle f(b) exists, but it does not provide any information on what that value is. This eliminates two of the four answer choices  - “The Intermediate Value Theorem can identify the value of \displaystyle f(c)=N that the function takes on as it passes from \displaystyle f(a) to \displaystyle f(b)” and “The Intermediate Value Theorem tells you how many times the function \displaystyle f repeats a value as it progresses from  \displaystyle f(a) to \displaystyle f(b).”

The Intermediate Value Theorem is useful because it can help identify when there are roots or zeros; an example of this is if a polynomial switches signs, Intermediate Value Theorem tells us there is a zero between those values.

From this, we can conclude that the correct answer is “The Intermediate Value Theorem states that somewhere between \displaystyle f(a) and \displaystyle f(b) there exists a value \displaystyle f(c)=N, with \displaystyle a< c< b.”

 

Example Question #113 : Limits And Continuity

In which interval does \displaystyle f(x)=x^3-5 have a root?

Possible Answers:

\displaystyle [0,1]

\displaystyle [1,2]

\displaystyle [-1,0]

\displaystyle [2,3]

Correct answer:

\displaystyle [1,2]

Explanation:

Graphing \displaystyle f(x)=x^3-5 on cartesian coordinates reveals that the function is continuous and crosses the \displaystyle x axis at a value within the interval \displaystyle [1,2]

 

Furthermore, setting \displaystyle x=1 produces a negative value for the function, while setting \displaystyle x=2 produces a positive value, as seen below:

\displaystyle f(x)=x^3-5

\displaystyle f(1)=(1)^3-5=1-5= -4

\displaystyle f(2)=(2)^3-5=8-5=3

 

Because the function is a polynomial, the function is continuous. By Intermediate Value Theorem, if the function changes signs within this interval, there must be a root present within the interval.

 

Example Question #111 : Calculus Ab

In which interval does the function \displaystyle f(x)=(x-1)^2-4 NOT necessarily have a root?

Possible Answers:

\displaystyle [2,4]

\displaystyle [-1.5,-0.5]

\displaystyle [4,5]

\displaystyle [-2,0]

Correct answer:

\displaystyle [4,5]

Explanation:

Because the function is a polynomial, the function is continuous. By Intermediate Value Theorem, if the function changes signs within this interval, there must be a root present within the interval. 

 

To apply Intermediate Value Theorem to the function \displaystyle f(x)=(x-1)^2-4, the function can be evaluated at each of the given bounds. 

 

For instance, if the function is evaluated at \displaystyle x=-2 and \displaystyle x=0, the following is obtained:

 \displaystyle f(x)=(x-1)^2-4

\displaystyle f(-2)=(-2-1)^2-4=9-4=5

\displaystyle f(0)=(0-1)^2-4=1-4= -3

 

For the intervals \displaystyle [-1.5,-0.5], \displaystyle [-2,0], and \displaystyle [2,4], there is a change in sign within the interval.

 

Because the function does not change in sign within the interval \displaystyle [4,5], we cannot conclude by Intermediate Value Theorem whether there is a root contained in the interval or not. Thus, \displaystyle [4,5] is the correct answer.

 

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