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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19k views

How to shift a line in a graph regardless of slope?

I want to shift a line right some points, regardless of its slope. For example, a vertical line will shift to the right by having its two y coordinates changed, [$(x_1, y_1+some number)$ $(x_2, y_2+...
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1answer
38 views

what is the slope of a line that 1) can rotate around a point, and 2) its reflection from a circle has a specific direction

I have a question. The figure of the problem: I have a line that intercepts a circle. the line (in vector form) has equation i + td , where i is the direction of the line, d is on point of the line ...
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1answer
1k views

Calculating error on slope of graph

I'm trying to find the rate of change and the error on that rate based on 7 measurements points and the assumption that the trend is linear. My calculations are below: $$ \begin{array}{|c|c|c|c|} \...
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13 views

Meaning of the Slope/Gradient of an Exponential Function Displayed on a Logarithmic Scale

I am currently dealing with data that is almost exponential. An exponential function shows as a line if plotted on a single logarithmic ordinate. To proof that it is of almost exponential behaviour I ...
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1answer
29 views

Explain how to find the equation of a line given the point (5,3) and it has a y-intercept of 13. [closed]

Explain how to find the equation of a line with the coordinate (5,3) and the y-intercept being 13.
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Restating my question about the form of the curves

I posted this question earlier. But because of my bad English, I could not state it very well, and two persons answered: arctan(x) and arctanh(x), but both of these answers are invalid. Now I am ...
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1answer
29 views

How do I put these two equations in slope intercept form? And how would I graph them?

$x+y=4$ $x-y=2$ Here is my work so far: $y=4-x$ $y=-2-x$
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1answer
1k views

Find slope tangent line to the graph of $f$ at $(\pi/2,\pi/2,0)$ in direction parallel to xy-plane.

Let $f(x,y)=x^2\sin(x+y)$ be any surface. Find the slope of the tangent line to the graph of $f$ at $(\pi/2,\pi/2,0)$ in the direction parallel to $xy$-plane. I am new to multivariable calculus and ...
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0answers
24 views

Help with rise and run of a graph that scales logarithmically

Bode Plot where the x domain is a scales my decade (log10) Hi guys i am just lost regarding this graph. I need to find the point $w<100$ where the graph starts to rise and the point $w>5000$ ...
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1answer
59 views

What does rate of change mean

I was wondering what does rate of change mean I know that the slope of position/time function for example, stands for the momentary velocity but I didn't understand why the velocity is $\frac{\text{...
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3answers
49 views

Slope of a quadratic function at its roots.

I've realised that every quadratic function with two real roots has slope $-1$ and $1$ at its roots. How do you prove that? Why $1/-1$? For example: $x^2 - 5x + 6$ has roots = $2$ and $3$. The ...
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1answer
53 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
64 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
43 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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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
14 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
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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1answer
38 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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0answers
23 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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2answers
1k views

Find the rate of change in a given direction of two variable function

The temperature of any given point on a plate is given by: T(x,y)=180e^(−(x^2/4)−(y^2/3)) I need to find the rate of change in temperature along the direction from (2,1) to (-1,-3). I have tried ...
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4answers
134 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
30 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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1answer
913 views

Barrow's Method For Slope of Tangent Line

I want to find the slop of the tangent line to the curve $2x^{3}y+4.5y-xy^{2}=8 $ at point (0,16/5) using Barrow's method. Here's what I've done. Substitute x with x+e and y with y+a $2(x^{3}+3ex^{2}+...
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1answer
79 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
69 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
42 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
51 views

Average Rate of Change of Functions

In here to find out the average rate of change -5 and 2 were used. But I'm wondering why, because in the interval these aren't included. Wouldn't it only be correct using -5 and -2 to solve the ...
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1answer
59 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
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
42 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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3answers
478 views

How to find the slope of curves at origin if the derivative becomes indeterminate

What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg-- What is the slope of the curve $x^3 + y^3= 3axy$ at origin and how ...
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2answers
34 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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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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26 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
29 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
20 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
38 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
31 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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33 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
33 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
23 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
123 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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