Questions tagged [slope]

For questions on finding or applying slope, a number that describes both the direction and the steepness of a line.

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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? ...
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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 = \...
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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 ...
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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 ...
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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-...
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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}...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $(...
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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 $y_0$?
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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 ...
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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 $f(x_k) = y_k$ ...
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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 ...
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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.
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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 ...
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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. ...
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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.
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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?
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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. ...
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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, ...
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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.
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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^...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ?
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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{\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $r$ ...
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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 = -...
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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 ...
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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 ...
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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 $...
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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 $...
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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?
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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 $y=2x+6$.
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...

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