Questions tagged [slope]
For questions on finding or applying slope, a number that describes both the direction and the steepness of a line.
276
questions
2
votes
1answer
60 views
Why does the limit give the exact value of the slope of the tangent?
This question has been already asked on this site many times but I haven't got a convincing answer and so some confusion lingers.
Why limits give us the exact value of the slope of the tangent line?
...
0
votes
0answers
25 views
The Method of Least Squares: uncertainty analysis for slope and intercept
Let be $m$ the slope and $b$ the intercept
With the Method of Least Squares these parameters are found to be
$$m = \dfrac{nS_{xy} - S_xS_y}{D}$$
$$b = \dfrac{S_yS_{xx} - S_xS_{xy}}{D}$$
with
$$S_x = \...
1
vote
1answer
35 views
Is this correct even if I get two different answers for slope-intercept form?
Writing an Equation for a Linear Function Given Two Points
If $f$ is a linear function, with $f(3)=−2$, and $f(8)=1$, find an equation for the function in slope-intercept form.
We can write the given ...
0
votes
0answers
16 views
Assuming Slope in Slope-Intercept Form [Clarification]
So when learning about slope-intercept form, the equation that was used looked like this: $y = 2x + 3$, it then displayed a graph where the slope was going up by two and across by one. What I need a ...
1
vote
2answers
12 views
Flat-top Gaussian distributions
I want to simulate microscope images (2D gray-scale pixel arrays) of fluorescent beads, which are tiny balls of polymer doped with fluorescent dye. I assume that an ideal image (neglecting out-of-...
2
votes
2answers
67 views
Find the angle between two tangents drawn from point $(0, -2)$ to the curve $y=x^2$
Find the angle between the two tangents drawn from point $(0, -2)$ to the curve $y=x^2$.
This is my attempt:
Let $P(\alpha, \beta)$ be a point on the curve.
$$\therefore \beta = \alpha^2$$
$$\frac{dy}...
0
votes
1answer
36 views
Find the Slope of the Tangent line with First Principle Method
Given the function $y =\sqrt x-1$, determine the slope of the tangent when x = 10. You must find the slope of the tangent using the method first principles.
I know how to find slope of a function but ...
0
votes
1answer
39 views
How to make a 2-d linear function using a third variable for the iterator? [closed]
Say, for example, you have the vector $\vec {PQ} = \langle8,4\rangle$. As we all learned in Algebra I, the "traditional" slope (y-units per x-unit) would be $\frac{4}{8}$, and the slope for ...
2
votes
0answers
22 views
Which rational slopes have angle bisectors with rational slopes?
This question was inspired
by the following question
in quora:
The lines
$y = 1/3 x$
and
$y = 13/9 x$
are drawn in the coordinate plane.
What is the slope of the line that bisects the angle these ...
0
votes
1answer
17 views
Slope of secant and tangent lines (supported by MVT)
For the function $f(x) = x^{1/3}$ on the interval $[1,8]$ find the point $(c,f(c))$ guaranteed by the Mean Value Theorem, at which the slope of the tangent line is equal to the slope of the secant ...
5
votes
4answers
132 views
If $|z|^2+\bar{A}z^2+A(\bar{z})^2+B\bar{z}+\bar{B}z+c=0$ represents a pair of intersecting lines… find the value of $|A|$.
If $|z|^2+\bar{A}z^2+A(\bar{z})^2+B\bar{z}+\bar{B}z+c=0$ represents a pair of intersecting lines with angle of intersection $'\theta'$ then find the value of |A|.
I tried using general equation of ...
3
votes
1answer
76 views
How can one point determine a unique straight line in differentiation?
I found a similar question and a beautiful answer here. However I'm not able to fully understand the answer and have a question on the selected answer at:
Consider all the lines going through point $(...
0
votes
1answer
41 views
Why don't we include $y$ approaches to $y_0$ as a limit? [closed]
How comes we only use $x$ approaches to $x_0$ when $y$ approaches $y_0$ is equally important. Why don't we include $y$ approaches $y_0$?
1
vote
1answer
57 views
What is the difference between average slope and Instant slope(Instantaneous Rate of Change) [closed]
I'm starting to learn calculus, and I'm getting confused about what average slope and instant slope(instantaneous rate of change)do and what they're differences are after looking at several sources on ...
0
votes
1answer
57 views
DId I find the right function f(x) = m*x+n?
I have the following task:
Let $(x_1, y_1)$ and $(x_2, y_2)$ be two points in the plane. We want to determine a
straight line given by the function $f$, i.e. $f(x) = mx + n$, such that $f(x_k) = y_k$
...
1
vote
1answer
47 views
Exponential equation from 2 points and slope
Problem: I am trying to calculate the formula for an exponential equation given $2 $points and the slope, but do not know the formula to do so.
What I have tried: With a quick google search, I found ...
1
vote
1answer
15 views
If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$?
If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$?
I searched the internet but I don’t understand. Please help! Thank you very much.
0
votes
1answer
40 views
Derivative of $e^x$ after geometric transformation
Inverse function of $f(x) = e^x$ is of course $f^{-1}(x) = \ln{x}.$ We have, by definition, $\frac{d}{dx}e^x = e^x$. In other words, $e^x$ in some sense describes slopes of tangent lines on a curve ...
1
vote
2answers
33 views
Derivative axiom
This is confusing me very much... Is there (rigorous) proof that slope of secant line "goes to" slope of tangent line on some point when $\Delta x \rightarrow 0$? This is actually not obvious at all.
...
0
votes
0answers
14 views
Is my definition of Average Rates Of Change and Instantaneous Rates Of Change correct?
Average rates of change: The slope of two points or secant.
Instantaneous rates of change: The slope of one point or tangent.
thanks.
2
votes
1answer
49 views
Geometric intuition for why the slope of the tangent at the y-intercept of $y=b^x$ is $\ln(b)$?
I understand how to prove that algebraically, but it's really amazing how the slope is exactly $\ln(b)$. My question is how can I develop a geometric feeling for it, or is it even possible to do so?
0
votes
0answers
11 views
How to turn asymptotic growth into constant growth?
In the function $f(x)=mx$, $x$ the value of $x$ is asymptotic as $m\to\infty$. I'm looking for a factor in terms of $m$ to multiply $f(x)$ by such that the growth in $x$ is constant as $m$ increases. ...
1
vote
0answers
17 views
Numerical solution for gradient(slope)
Abstract
I have the next equation to find a force, for my problem:
$$U=-\int \vec{m}\small{(x)}\times \vec{B}(x)dV$$
$$\vec{F}=-\nabla U$$
Considering 3-dimensional space with x,y,z coordinates, ...
0
votes
2answers
25 views
Find the missing rise that makes these lines perpendicular. [closed]
The slopes of two lines are
$m_1 = -3$ and $m_2 = k/4$.
Find the value of k that makes these lines perpendicular.
0
votes
0answers
25 views
Use the $x(t)$ functions describing the motion of the two cars to predict where they will meet.
We have Car A and B, and I have the position of each car:
Car A:
$A= 1.374E2 *10^2$, $B= -2.719E2 *10^2$
$x= 1.374*10^2 -2.719*10^2...(1)$
Car B:
$A=6.473E2= 6.473*10^2, B=-4.402E2= -4.402*10^...
0
votes
2answers
42 views
How do I determine if 3 points fall on a straight line?
$1.\;A(0,0,0),\,B(9,−4,3),\,C(−36,16,−12)\\
2.\;D(9,−4,3),\,E(10,−2,6),\,F(14,6,18)\\
3.\;G(−1,0,1),\,H(3,9,10),\,I(8,27,28)$
I want to determine if these points fall on a straight line.
From my ...
0
votes
1answer
28 views
How to find formula of line extruding from intersection point from two lines [closed]
wasn't exactly sure how to word this in a quick question title, but say I have the following:
Meaning, I have two lines (or line segments) with known slopes and start / end verticies, and one ...
0
votes
0answers
28 views
Intuition of product rule using graphs and slopes
I have seen formal proof of the product rule ($(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$) and one with rectangles and areas - explanation seems reasonable. But, is there direct and intuitive proof using ...
0
votes
0answers
18 views
Why does a kink sometimes ruin convexity of a set and other times not?
Specifically, I have the following in mind:
Consider the positive quadrant as the domain (i.e. $(x,y)$ with $x,y\geq 0 $).
Consider a downward sloping line that divides this domain into two sets (the ...
1
vote
2answers
37 views
Calculus A Level Line Tangent to Circle
How can you find values of $k$ such that $y = kx + 1$ is tangent to the circle $(y-1)^2 + (x-5)^2 = 9 $?
I first rewrote the circle equation in terms of y:
$$ (y-1)^2 = -(x-5)^2 + 9 \\y-1 = \pm\...
-1
votes
1answer
30 views
How and why is Rise/Run gives the inclination of the line (or slope)? [closed]
How and why is Rise/Run gives the inclination of the line (or slope)?
How to understand the theoratical concept behind it ?
0
votes
0answers
30 views
Pulling out terms in solving the covariance of the slope and intercept in linear regression
I want find the covariance of the estimates $\hat{\beta_0}$ and $\hat{\beta_1}$. There are many answers such as this one that give the answer as
\begin{align*}
\operatorname{Cov}(\hat{\beta_0}, \hat{\...
0
votes
2answers
31 views
Converting a semilog slope to a log-log slope
I'm working on species area relationships. Basically, the more area you have, the higher the species richness. It's often described by a power function:
$S=c A^z$
The contemporary way to use this ...
1
vote
0answers
21 views
MIT OpenCourseware Single Variable Calculus, Question about Derivatives in Problem 1C) 6
In the homework for MIT Open Courseware Single Variable Calculus, one of the questions involves drawing the derivatives of functions. I have included the picture from the answer key of the functions ...
2
votes
3answers
115 views
How to find the slope of $y.\ln x = x.\ln y$ at $x = e$?
Let's say we have the following equation:-$$y.\ln x=x.\ln y$$
After graphing the equation on desmos (which included, not surprisingly, the line $y=x$), I realised that the equation has a slope of 1 ...
0
votes
1answer
30 views
What is the angle (in degrees) that a line with slope $m$ makes with the x-axis (x>=0)
I have tried this many times, but all I have figured out is that if one line has slope $a$ and another line has slope $b$ then the ratio of the angles made with the y-axis is the same as the ratio of ...
0
votes
1answer
29 views
Slope of a Line Relative to $r$ and $\theta$ Basis Vectors
I was reading Purcell (a well-known physics textbook for E&M) and I stumbled upon something which bothered me. Purcell was trying to explain how to find the slope of a line with respect to the $r$ ...
-1
votes
1answer
28 views
determining accurate points on x/y graph with fractions
I have the following 2 points and need to find other points along this line and the y- intercept.
$$\left(-\frac13, \frac54\right), \left(\frac12, \frac34\right)$$
I have determined the slope $m = -...
3
votes
2answers
396 views
Why derivative is a slope?
The change of $Y$ per $X$ is slope.
And some say the change of slope per $X$ is derivative.
So it is like slope of a slope!
But slopes are always numbers like the slope of $2x$ is $2$. But derivates ...
1
vote
3answers
50 views
Find the equation of a parabola tangent to an exterior circle/arc
I am trying to find the equation of a parabola that is tangent to an exterior circle (this is for designing a bell-shaped nozzle). I know the point (0.055, 0.9) lies on the parabola and also that the ...
2
votes
3answers
575 views
Find locus of $\Delta ABC$ centroid with orthocentre at origin and side slopes 2, 3 and 5
Let $ABC$ be a triangle with slopes of the sides $AB$, $BC$, $CA$ are $2,3,5$ respectively. Given origin is the orthocentre of the triangle $ABC$. Then find the locus of the centroid of the triangle $...
2
votes
1answer
39 views
What is the notation $f'(x^+) $ and $f'(x^-)$?
Differentiability of a function at a point $x$ is confirmed when
$\lim_{h\to0} \frac{f(x+h)-f(x)}h$ exists and is finite.
But in some textbooks it is noted that a function is differentiable if $...
0
votes
1answer
27 views
Show that $f(x) = \frac{1}{x\ln x[\ln(\ln x)]^{\frac23}}$ is decreasing for all $x > 3$
Show that $$f(x) = \frac{1}{x\ln x[\ln(\ln x)]^{\frac23}}$$ is decreasing $\forall x > 3$.
How can I show this?
0
votes
1answer
30 views
Find a tangent line on a function equal to a specific slope [closed]
Find the tangent line on $f(x)=x^2$ which runs parallel to the slope of $y=2x+6$.
0
votes
2answers
53 views
Regarding the calculation of the slope of a tangent.
I have recently had some of my first lessons in calculus. We've learned to use the well-known formula for the slope of a tangent:
$$m_t=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$
When working with this ...
0
votes
2answers
43 views
How to interpret the slope in this chart?
I have a chart that was generated from a linear regression. It looks like this:
I see that it's slope is slightly down. The slope I get from the linear regression, however is -1.3081201334816588E-9
...
0
votes
0answers
22 views
Jordan-Hölder sequence for $\mu$-semistable sheaves.
Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class.
I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
0
votes
1answer
22 views
Finding the rate of change of an algebra equation
I'm trying to help my daughter with her math homework and after scouring google and websites I find I truly do not understand how to complete this even after going several different websites.
The ...
0
votes
3answers
25 views
Oblique asymptotes of non rational, non function
I'm given the equation:
$6x^2y+xy^2-2x^3=0$
and I'd like to know the slope of the two lines it forms.
While graphing a more general form: $ax^2y+bxy^2-cx^3=k$, I've noticed that if you set the ...
0
votes
0answers
246 views
Slope field and phase portrait of ODEs intuition
I'm trying to develop an intuition for slope fields and phase planes/portraits that we use to analyze the stability of equilibrium solutions of ODEs.
Given a first order ODE like
$y' = f(t,y)$
We ...
