0
votes
1answer
42 views

Limit of a function containing square root.

Q. $\lim _{x\to 0}\frac{\left(\sqrt{1-cos2x}\right)}{x}$ We can write this function as $\lim _{x\to 0}\frac{\left(\sqrt{2sin^2x}\right)}{x}$. Algebraically we have ...
0
votes
1answer
24 views

Log arithmic Equation - Graph curved line

I'm recreating the graph picture below with equations. Using the online graphing tool "Desmos": These are all the equations I have done so far, with there restrictions top stop at specific points. ...
0
votes
1answer
46 views

Graphing picture equations - Curve Lines

I'm basically trying to recreate the graph picture below. Using a online graphing tool "Desmos": I managed to create the equations for the straight lines and circles for the sunset picture. ...
0
votes
1answer
22 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
1
vote
1answer
48 views

Function - Main Features?

I understand how to draw this function, but what does it mean by main features? any examples for the question below? Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = ...
5
votes
2answers
49 views

Intuitive characterization of the graph of a twice differentiable function

In high school textbooks, the following characterizations are often found: A function is continuous if its graph can be drawn without lifting the pencil. and A function is differentiable if ...
0
votes
2answers
79 views

Finding roots with seemingly no algebraic way

I have a graph of: $$y = \frac{x^3 + 2x^2 - 4}{x^2}$$ and I have to find the x-intercept. So I have the equation $(x^2)(x+2)-4 = 0$ And then I don't know what to do. Not sure if we can use ...
3
votes
2answers
89 views

Behaviour of the function $\ln(1+ x^2)$

Thus function has derivative equal to: $\frac{2x}{1+x^2}$. This indicates that it will flatten out while approaching infinity, ie, should have an asymptote. Yet, the function does not have any real ...
0
votes
1answer
24 views

Is the following graph having two local minima

https://www.desmos.com/calculator/abuvb1zdkb I think yes, the main question i think is of the definition of neighbourhood For a function with domain $(-\infty, -3)\cup (3, \infty)$ $ $ Is -3 in ...
1
vote
0answers
77 views

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points?

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points? I don't quite the understand the solution which is in ...
0
votes
1answer
28 views

Find the slant asymptote for $x= \dfrac{3t}{1+t^3}$, $y= \dfrac{3t^2}{1+t^3}$

I am geting $y=-x$, while the answer $y=-x-1$ What i did is to write $y = tx$ and as $t$ goes to $-1$. $x, y$ go to $\infty$, hence the asymptote is $y=-x$ Where did i go wrong.
3
votes
2answers
270 views

What is a French curve, as mentioned by Feynman?

I'm reading "Surely You're Joking, Mr. Feynman!", he says: I often liked to play tricks on people when I was at MIT. One time, in mechanical drawing class, some joker picked up a French curve (a ...
0
votes
1answer
39 views

What is the graph of $y = \sin n$ and why is it different from the graph of $y = \sin x$?

I have downloaded a book about Calculus from MIT OCW. In that book, there is a section "A Thousand points of Light". (You can download the relevant section from here.) In that section, it is written ...
0
votes
1answer
26 views

Rewrite the following surface so that I can graph it.

$z = \dfrac{1+x^2}{1+y^2}$ $ $ I want the part of the surface above the square $|x|+|y|\leq 1$ $ $ OR we can write this square as $-y<x<y$ and $-1<x<-1$ $ $ I have spent hours trying ...
2
votes
2answers
31 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
0
votes
4answers
46 views

Increasing, continuous, concave downward function normalised between 0.5 and 1

What would be a good function which is increasing, continuous, concave downward with $$\lim_{x \to 0} f(x) = 0.5$$ and $$\lim_{x \to \infty} f(x) = 1.$$ It should be concave downward whose ...
1
vote
1answer
21 views

Plotting the intersection of multiple surfaces with WolframAlpha

I want to plot the intersection of two surfaces like in this post. But if I enter the much simplified expression ContourPlot3D[{x^2 + y^2 + z^2 - 4=0, xy=1}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] ...
1
vote
1answer
43 views

How to draw the graphs for these functions?

Could I please be helped with the graphs for the following functions: $$y=\lceil \tan x \rceil, \quad y=\tan (\lceil x \rceil), \quad y=\lceil \tan (\lceil x \rceil) \rceil$$ I have been able to ...
0
votes
0answers
21 views

Mathematical/graphical implications of representing double integral of changing acceleration (which gives displacement) on a 3d plane.

Lets assume a function a = |sin t| for acceleration over time. If we integrate it, we get instantaneous velocity. I have been told that integrating a function provides me with the area under the ...
0
votes
1answer
38 views

Integrating function of acceleration to obtain displacement

Lets assume a function a = |sin t| for acceleration over time. If we integrate it, we get instantaneous velocity. Now i have taken a limit for time. How will this graph look like? I have been told ...
2
votes
2answers
36 views

Draw a function $g$ such that $g$ is defined on the interval $[-9,9]$

Sketch a function $g$ such that $g$ is defined on the interval $[-9,9]$ and satisfies the following properties: a) $g(-5) = 1$, but $\lim_{x \to -5} g(x)$ does not exist. b) $\lim_{x \to 1} g(x) = ...
0
votes
1answer
32 views

how to find the foci, directrix, center of a polar conic section. ($r=\frac{4}{5-4sin\theta} $)

I've been trying to figure this out for a bit and haven't found an answer. the equation is this: $r=\frac{4}{5-4sin\theta} $ I know I need to match this up to a conic graph so I divide top and ...
0
votes
0answers
28 views

I was wondering if there was a way to plot a vector field and a 3d surface in the same window

I want to plot the vector field $$\textbf{F}(x, y, z) = \sin(xyz)\textbf{ i }+ x^2y\textbf{ j }+ z^2e^{x/5}\textbf{ k }$$ And the surface (part of elliptic cylinder) $$\textbf{r}(u, v) = u\textbf{ i ...
1
vote
3answers
54 views

Why rotating a function around line $y=x$ gives an inverse of this function?

So I'm trying to read through a book on calculus on my own and there is a statement that if we have a graph of some function $y=f(x)$ and this is an injective function, then rotating it around the ...
4
votes
1answer
47 views

Speediness and correctness when graphing by hand .

First of all thank you for visiting this question! I believe it's a pretty simple problem but get's kinda hairy and time consuming on each step as I have done it, so my question (the one you are here ...
0
votes
4answers
82 views

Multiple choice question on rates of change (or so I thought)

If I were to find the resistance of the component (see image below), I would either find the equation of the curve and use differentiation or I'd draw a tangent at $V_2$ and then find the reciprocal ...
1
vote
1answer
28 views

Find Sin Function from points?

I have these points and I need to find the sine function can anyone please show me how to do this step by step? Or just give me pointers on where to start and what the equation actually is? (0,2.8) ...
1
vote
1answer
67 views

Describe the graph of f if the graph of its integral its given

Describe the graph of $f$ if the graph of its integral $g(t) = \int_{0}^{t} f(s) ds $ is: graphic of g graphic of f I analyze the derivative and the sign of the derivative and try to find ...
0
votes
2answers
29 views

$y = xe^{−1/|x|}$ for $−∞ < x < ∞$

(a) Let $f(x) = x − xe^{−1/x}, x > 0$. Show that f(x) is an increasing function on $(0,∞),$ and $\lim_{x→∞} f(x) = 1$. (b) Using part (a) and calculus, sketch the graphs of $y = x−1, y = x, y = x ...
2
votes
0answers
50 views

$x^{y}=y^{x}$ Intersection Question

I'm looking at this problem, $x^{y}=y^{x}$ and the question is to find the point of intersection of the two curves that form a solution set. I understand how to find the point of intersection using a ...
5
votes
5answers
123 views

Points on $(x^2 + y^2)^2 = 2x^2 - 2y^2$ with slope of $1$

Let the curve in the plane defined by the equation: $(x^2 + y^2)^2 = 2x^2 - 2y^2$ How can i graph the curve in the plane and determine the points of the curve where $\frac{dy}{dx} = 1$. My work: ...
2
votes
4answers
98 views

Understanding the graph for $x^y = y^x$

I graphed $x^y=y^x$ and it is a union of the line y=x, with some other curve. So my first question is, how do I derive that other curve? My next question is, why don't I get the same graph when I ...
0
votes
3answers
65 views

Find the locus of points

Find the locus of points, the distance between them and the point $(2,1)$ is equal to the distance between them and the straight line $4x = 3y$ I know that it is the definition of a parabola But I ...
0
votes
2answers
44 views

creating graphs in wolfram alpha

I am trying to plot the function $$\max(x-40,0)$$ in Wolfram|Alpha, but I can't figure out how. What Wolfram|Alpha query URL should I use to produce this graph? Or, what similar online program could ...
1
vote
1answer
32 views

description of the function whose graph corresponds to Figure

Consider f be a real continuous function , $f(0) = 0$ , and whose graph has the form shown in the figure: a) How can a give description of the function whose graph corresponds to Figure. b) Sketch ...
0
votes
0answers
17 views

Trouble calculating surface area

I am trying to calculate the surface area of a solid of revolution of f(x) about the x axis for the interval [1,6] $f(x) = \begin{cases} 1 & 1 \leq x< 2\\ 1/2 & 2 \leq x< 3\\ . ...
0
votes
1answer
29 views

Finding surface area of revolution

Can anyone help me with finding the surface area of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having ...
1
vote
1answer
50 views

Finding volume of the solid of revolution?

Can anyone help me with finding the volume of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having ...
1
vote
1answer
30 views

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$?

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$? I see that is should be $-\dfrac{\pi}{2}\leq \ln(x\,\arccos{y}) \leq \dfrac{\pi}{2}$ and then $e^{-\dfrac{\pi}{2}} \leq x*\arccos(y) \leq ...
0
votes
3answers
38 views

What is the domain of $z=\arcsin\dfrac{x}{y}$?

I get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well, ...
0
votes
0answers
20 views

Locus of a moving point

Let $f(x),g(x)$ be Differentiable functions in their respective domains. Let $m(x)$ be a function whose perpendicular distance from $f(x)$ is always $g(x)$.Then, whats the equation of $m(x)$ in terms ...
3
votes
1answer
57 views

The continuity of a function at a point x0

Choose correct options , more than one may be correct . (1) $$ \textrm{The function defined by } \begin{cases} f(x)=\cos\left(\dfrac{1}{x}\right) & x\neq 0\\ f(0)=0 & \\ \end{cases} ...
0
votes
2answers
41 views

How to sketch a function such as r(t) = (t^2)i + (t^3)j

How can I sketch this function by hand? I am not even sure what I should be expecting,.
0
votes
2answers
30 views

Tangents and differentiability

Was just going through my single variable calculus notes recently when I came across this interesting article on the relation between differentiabilty of a function. I missed some of the points my ...
0
votes
1answer
49 views

setting up the equation of area between two polar curves

I have to find the area inside $r=2\cos x$ and outside $r=1.$ Can someone please check if I set up my equation correctly? I did: $$2\left(\frac12\int\limits_0^\pi 2\cos^2 x\ ...
0
votes
0answers
214 views

finding area of one loop of a lemniscate

I have to find the area of one loop of the equation r^2=4cos(2x) (I'm using x as a substitute for theta). I'm getting confused because the answer is 2 but I keep getting 0. I tried to type my ...
0
votes
1answer
23 views

Graphing a function?

How can you have a function on the $x$ and $y$ axis? I thought graphing functions worked like this: You have a variable on one of the axis, and a function of the variable on the other axis. How can ...
3
votes
3answers
91 views

Calculus: tangents and limits

For 1), I know the graph has a tangent but I don't know how to explain. For 2), is the answer 0? Would anyone mind giving me the steps? For 3) and 4), I really have no idea.
0
votes
2answers
2k views

How can you find the x-coordinate of the inflection point of the graphs of f'(x) and f''(x)?

So I understand how to find the inflection points for the graph of f(x). But basically, I have been shown a graph of an example function f(x) and asked the state the inflection points of the graph. ...
0
votes
2answers
47 views

What will be it's graph?

What will be the graph of $y=2x +\sin x $ and $y=x \sin x$ and what's the method to graph functions of this type.