Are there errors in these multiple choice questions?

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hndalama

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I think I have spotted some errors in a few questions on a test I did and I want to verify that I am right before I enquire about them. They are just 3 questions.

1.) At what value of x does the absolute maximum of f(x)=−x2+1 occur on the interval [−2,4]?

well the absolute maximum of the function occurs at x=0 which is within the interval. the options are
a) left endpoint b)an interior point c) right endpoint d) cannot be determined

I picked b) interior point but my answer has been marked wrong

2.) Which of the following is not a criterion for Rolle’s Theorem to guarantee the existence of a local extreme point on an interval [a,b] for a function f(x)?

a. f(x) is continuous on the interval.b. f(a)=f(b)=0
c. f(x) is differentiable on the interval.
d. f(x) is concave up on the interval
e. Both C and D

Here I think b and d are both 'not a criterion,' b is wrong because rolles theorem applies even when f(a) and f(b) do not equal zero. and d is wrong because there would be a local maximum if f(x) was concave down.

3.) Find the length of the curve over the interval [0, 1].

using the integral formula for length of a curve I calculate the integral \(\displaystyle \int_{0}^{1}\sqrt{1+81x}dx\)
the antiderivative is 2/3(1+81x)​3/2
and that solves to \(\displaystyle 2/3(82\sqrt{82}-1)\)

the options given are

\(\displaystyle 2/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-2)\)
\(\displaystyle 2/243(82\sqrt{82}-2)\)


I have looked at this again and again and I have no idea where the 243 is coming from.

That's all the questions, please let me know if I am mistaken on any of them. Thank you
 
Last edited:
hndalama said:
I think I have spotted some errors in a few questions on a test I did and I want to verify that I am right before I enquire about them. They are just 3 questions.

1.) At what value of x does the absolute maximum of f(x)=−x2+1 occur on the interval [−2,4]?

well the absolute maximum of the function occurs at x=0 which is within the interval. the options are
a) left endpoint b)an interior point c) right endpoint d) cannot be determined

I picked b) interior point but my answer has been marked wrong

2.) Which of the following is not a criterion for Rolle’s Theorem to guarantee the existence of a local extreme point on an interval [a,b] for a function f(x)?

a. f(x) is continuous on the interval.b. f(a)=f(b)=0
c. f(x) is differentiable on the interval.
d. f(x) is concave up on the interval
e. Both C and D

Here I think b and d are both 'not a criterion,' b is wrong because rolles theorem applies even when f(a) and f(b) do not equal zero. and d is wrong because there would be a local maximum if f(x) was concave down.

3.) Find the length of the curve over the interval [0, 1].

using the integral formula for length of a curve I calculate the integral \(\displaystyle \int_{0}^{1}\sqrt{1+81x} dx \)
the antiderivative is 2/3(1+81x)​3/2
and that solves to \(\displaystyle 2/3(82\sqrt{82}-1)\)

the options given are

\(\displaystyle 2/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-2)\)
\(\displaystyle 2/243(82\sqrt{82}-2)\)


I have looked at this again and again and I have no idea where the 243 is coming from.

That's all the questions, please let me know if I am mistaken on any of them. Thank you
Click to expand...
For #3
\(\displaystyle \int_{0}^{1}\sqrt{1+81x} dx \)

let u = 1 + 81x → du = 81* dx

\(\displaystyle = \ \frac{1}{81}\int_{1}^{82}\sqrt{u} du \)

\(\displaystyle = \ \frac{2}{3 * 81}\left [(82)^{\frac{3}{2}} - 1 \right ] \)

do you see 3 * 81 = 243
 
Subhotosh Khan said:
For #3
\(\displaystyle \int_{0}^{1}\sqrt{1+81x} dx \)

let u = 1 + 81x → du = 81* dx

\(\displaystyle = \ \frac{1}{81}\int_{1}^{82}\sqrt{u} du \)

\(\displaystyle = \ \frac{2}{3 * 81}\left [(82)^{\frac{3}{2}} - 1 \right ] \)

do you see 3 * 81 = 243
Click to expand...

Oh yes, I see my mistake now. thank you

What about the other 2, do you think I have made a mistake there as well?
 
hndalama said:
I think I have spotted some errors in a few questions on a test I did and I want to verify that I am right before I enquire about them. They are just 3 questions.

1.) At what value of x does the absolute maximum of f(x)=−x2+1 occur on the interval [−2,4]?

well the absolute maximum of the function occurs at x=0 which is within the interval. the options are
a) left endpoint b)an interior point c) right endpoint d) cannot be determined

Calculate:

f(-2) = ??

f(-4) = ??

f(0) = ??

Now think..... what was asked originally?

I picked b) interior point but my answer has been marked wrong

2.) Which of the following is not a criterion for Rolle’s Theorem to guarantee the existence of a local extreme point on an interval [a,b] for a function f(x)?

a. f(x) is continuous on the interval.b. f(a)=f(b)=0
c. f(x) is differentiable on the interval.
d. f(x) is concave up on the interval
e. Both C and D

Here I think b and d are both 'not a criterion,' b is wrong because rolles theorem applies even when f(a) and f(b) do not equal zero. and d is wrong because there would be a local maximum if f(x) was concave down.

3.) Find the length of the curve over the interval [0, 1].

using the integral formula for length of a curve I calculate the integral \(\displaystyle \int_{0}^{1}\sqrt{1+81x}dx\)
the antiderivative is 2/3(1+81x)​3/2
and that solves to \(\displaystyle 2/3(82\sqrt{82}-1)\)

the options given are

\(\displaystyle 2/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-1)\)
\(\displaystyle 4/243(82\sqrt{82}-2)\)
\(\displaystyle 2/243(82\sqrt{82}-2)\)


I have looked at this again and again and I have no idea where the 243 is coming from.

That's all the questions, please let me know if I am mistaken on any of them. Thank you
Click to expand...
.
 
f(x)= −x2+1
f(-2)=-3
f(-4) = -15
f(0) = 1

the question is asking for the x value that outputs the 'absolute maximum' on that interval so x=0 is the value where the absolute maximum occurs. Is something wrong?
 
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