Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in $(-2, -\frac{2}{3})$ then find $a$

If $f(x) = x^3 + 4x^2 + ax + 1$ is a monotonically decreasing function of $x$ in the largest possible interval$(-2, -\frac{2}{3})$ then find $a$. My work: $$f'(x)=3x^2+8x+a$$ For it to be ...
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How to simplify trigonometric functions with having higher multiples of $x$ if the function is complex?

$$\int \frac{(\cos 9x + \cos6x)}{2 \cos 5x - 1} dx$$ I know that it simplifies to $\cos x + \cos 4x$ but I have no idea how to do that. I tried expanding $\cos 9x$ and $\cos 6x$ by using the ...
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How to solve the inverse function of $f(x) = x^a + x^b$ [on hold]

I don't even know where to start. I have tried everything. I even googled it.
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What will be the value of x? [on hold]

Find $\lim_{x\to \frac 52} \lfloor x\rfloor$ , where $\lfloor\,\,\rfloor$ denotes the greatest integer function , Please do not close this question or downvote .
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Integral of determinant of Jacobian of a $C^{1}$ application.

Let $f \in C^{1}(\mathbb{R}^{n}; \mathbb{R}^{n})$, suppose that exists $r>0$ such that $f(x)=0$ if $|x|>r$. Prove that exists $k>0$ such that $\int_{\overline{B_{k}(0)}} det Jf(x) dx=0$. My ...
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Difficulty setting up an iterated integral

I am trying to integrate the function $\frac{1}{\sqrt{2y-y^2}}$ over the region in the first quadrant bounded by $x^2=4-2y$. Given that this region is between bounded by an convex parabola and in the ...
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Find the volume of the solid generated by rotating about the x-axis the region bounded by the curves

I'm new to this site, and this will be my last question post. My boyfriend has a few problems that he's unable to complete because of a family emergency and I have decided to try and help him. His ...
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Find the depth of the propane in the tank when it is filled to one-quarter of the tank's volume.

I'm new to this site. My boyfriend has some calculus problems that he's unable to complete due to a family emergency, and so I am trying to help him. His professor hasn't emailed him back about ...
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Finding the volume of a solid generated by revolving around the line y=6

I'm new to this site. My boyfriend is unable to complete these problems due to a family emergency and I was going to try to help him because they are due in a few hours and his professor hasn't ...
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Reading an article that takes the cross product of a vector and time.

I have this equation in the form: $$V(t)=ϕV(t−1)+r×t+c+\epsilon$$ Why is $r$ crossed with $t$. The variable $t$, in the article (shown below) is time which I believe is a scalar. I suppose time ...
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volume of solid around y axis rotation

I am working on this problem and need some help figuring it out. I will give the problem and then give the ways i tried to solve it: Find the volume of the solid generated by revolving the region ...
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How is the class related to derivability?

Good evening to everyone. I have a question where they require me to find the derivability. After I read the answer sheet I saw that the function has the class $C^1$. How is the class related to ...
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How long is 1 turn of conical helix? [on hold]

The cone has diameter 120mm and angle of slant is 31°. What is the lengh of 1 turn of conical helix (angle 8°)?
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Is L a linear transformation?

I have to prove is L is a linear transformation on the field $P_3(R)$, if it is then I'd have to find the matrix of the linear transformation from the standard base vectors $p(1),p(x),p(x^2),p(x^3)$. ...