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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13
votes
1answer
163 views

Understanding the slope of a line as a rate of change

I thought I was confident about what is meant by the slope of a line and how it relates to a rate of change, but I'm having doubts and I'm hoping that someone will be able to help me clear them up. ...
9
votes
1answer
127 views

If $\sigma: \mathbb{R}\to \mathbb{R}^2$ is a function that spirals,goes to infinity and repeats itself, then is $\sigma$ non-injetive?

Let $\sigma:\mathbb{R}\to \mathbb{R}^2$ be a smooth function such that $$\frac{\text{d}\sigma}{\text{d}t}(s) \neq 0, \quad \forall s \in \mathbb{R},$$ and $$\sigma(t+n) = \sigma(n) +\sigma(t),\ \...
5
votes
5answers
9k views

How to find x and y coordinates based on the given distance?

The problem says: Find the point (coordinates $(x,y)=~?$) which is symmetrical to the point $(4,-2) $ considering the given equation $y=2x-3$ I have found the perpendicular line-slope $y=-~\frac{1}{2}...
5
votes
6answers
1k views

Why limits give us the exact value of the slope of the tangent line?

Limits tell us how functions behave at $x\to a$, not how they behave at $x = a$. However, in limits we plug $x = a$ as an approximation of $x\to a$, so: why the limits give us the exact value of slope ...
5
votes
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 ...
5
votes
2answers
499 views

What is the equation for an ellipse given 3 points and the tangent line at those points?

Does anyone know how to find this? I know I have enough information to uniquely generate an ellipse (you only need five points for a conic, I have three plus the slopes or tangents at those points ...
4
votes
1answer
432 views

(Calculus) Derivative Thinking Question

Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it. The ...
4
votes
2answers
26k views

Find slope using X-Intercept and Y-intercept

My mathematics teacher explained how to find slope using 2 different points but I was never taught how to use the X-Intercept and the Y-Intercept to find the slope. Example question: What is the ...
4
votes
1answer
231 views

Find Slope Of Line Drawing A Circle

I am programming a semi-circle (and it can also be a circle) that animates to full and empty like so: I am having a hard time calculating is the slant on the line. What I would like to do is have a ...
3
votes
1answer
315 views

What characterizes a tangent line?

If the traditional way to define the tangent line to a curve $f(x)$ through the point say $(a , f(a))$ is: ( the tangent line through the point $(a ,f(a))$ is the line that passes through this point ...
3
votes
2answers
419 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 ...
3
votes
6answers
391 views

Definition of a slope of a straight line [closed]

I am learning about the straight line and its slope was defined as "rise over run". Why not "run over rise"? Was "rise over run" chosen arbitrarily or it was derived based on some physical ...
3
votes
3answers
145 views

How can I find m using the discriminant?

There is a curve $$y=2x^2-4x+8$$ and a line $$y=mx$$. I have to find the values of m for which the line is tangent to the curve. I made them equal, then put everything to one side: $$2x^2-4x+8-mx=0$$ $...
3
votes
3answers
823 views

Drawing Straight line graphs without a table of values grade 7

I am a a student and I am having difficulty with answering this question. I keep getting the answer wrong. Please may I have a step-by-step solution to this question so that I won't have difficulties ...
3
votes
2answers
72 views

Is there a mathematical term for telling if $|m|$ is greater than or less than one?

A slope can be positive, negative, zero or undefined, but what about the magnitude of the slope? It seems useful to recognize if the slope is steep (greater than $1$) or gentle (less than $1$) as much ...
3
votes
1answer
77 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 $(...
3
votes
1answer
131 views

Rational Coordinates with two intersecting lines. [duplicate]

If there are two 2D lines with rational slopes that intersect, must the intersection point have rational co-ordinates? I'm having a hard time wrapping my head around this and not sure how you prove ...
3
votes
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 ...
3
votes
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 ...
3
votes
1answer
164 views

Quadratics: Intuitive relation between discriminant and derivative at roots

While working with quadratics that have real roots, I realized an interesting fact: The slope of a quadratic at its roots is equal to $\pm \sqrt{D}$ where $D=b^2-4ac$ Proof: $$f(x) = ax^2 + bx +c$$...
3
votes
1answer
25 views

Alternative Proof to Proving Inequality Involving Slopes?

I was wondering if there was any other way of proving: Is this idea regarding slope true and how do you prove it? As I feel like I wouldn't be to catch that on say, a test, I was hoping if there is ...
2
votes
3answers
184 views

Why $\infty×0=-1$ from multiplication of two slopes of two lines perpendicular to each other and how do we define infinity?

Here is given $A(x_1,y_1), B(x_1,y_2), C(x_2,y_3)$ and $D(x_3,y_3)$. I have recently read that, multiplication of two perpendicular lines is always $-1$. From the above graph, the slope of $AB, m_1 = ...
2
votes
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 ...
2
votes
3answers
35 views

Finding slope m of tangent to curve

been years since I took calculus and am currently struggling with how to properly work out the following: Using the tanget line slope formula: My understanding is that I would need to find the ...
2
votes
2answers
73 views

Intercept of a Line problem [closed]

How would one find the x-intercept and y-intercept of the line with the equation $x-2y=0$?
2
votes
1answer
56 views

Finding the normal to a curve of an implicit function using differentiation

The question asks to find the tangent and normal to the curve of the following equation at the point ($\sqrt3, 2$): $$x^2-\sqrt{3}xy+2y^2=5 $$ I began by finding the first derivative of this, ...
2
votes
2answers
33 views

Calculating tangent line at $t=2$

Calculate the tangent line at $t=2$ of the curve $$x(t)=t^2+2t+4,\ y(t)=1+te^t$$ How would you go about determining this? So far I have $$x'(t)=2t + 2\\ y'(t) = te^t+e^t$$ Where do I go from here ...
2
votes
1answer
5k views

Is the slope of a vertical line infinity or undefined?

By slope I mean derivative. Is something being infinity the same thing as being undefined? Is the slope of a vertical line negative infinity or positive infinity?
2
votes
1answer
2k views

Find the equation of the tangent to the parabola $y=x^2$, if the x-intercept of the tangent is 2

I'm trying to solve this problem: Find the equation of the tangent to the parabola $y=x^2$. If the x-intercept of the tangent is 2. All what I can think of is finding the slope which is $...
2
votes
1answer
2k views

Changing the side of a triangle without changing area?

$\triangle ABC$ has vertices $A=(8,2)$, $B=(0,6)$ and $C=(-3,2)$. Point $C$ can be moved along a certain line with points $A$ and $B$ remaining stationary so that the area of $ABC$ will not change? ...
2
votes
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? ...
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}...
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 $...
2
votes
1answer
640 views

Does an endpoint of a function have a slope?

Say I have the following function which is defined for $0 \leq x \leq 1$: If you would derive this function and substitute $y = 0$, you would get $x = 0.5$ and $x = 0.75$ for the respective as the ...
2
votes
1answer
2k views

Is there a correct mathematical term for the inverse of the slope?

Linear graphs all have a slope that can be calculated by deviding the progress on the y-axis by the progress on the x-axis. Is there a correct term to refer to the inverse of the slope, that means the ...
2
votes
2answers
66 views

Concept of the slope

Having difficulties understanding the concept of the slope: Suppose we have $f(x)=x^2$, its derivative $f'(x)=2x$ At $x=10$, $f(x)=100$ and $f'(x)=20$. So the rate of change of the function at $(x,y)...
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?
2
votes
3answers
594 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
104 views

The slope of $nx\ \%\ m$

(There is a follow-up question at MO.) Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e. $$x \equiv x\ \%\ m\pmod{m}$$ Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e. $$\mu^n_m(...
2
votes
1answer
708 views

How to find tangents to curves at points with undefined derivatives

I will explain my question with the help of an example. We need to find the tangent at origin to the curve $$x^3 + y^3 =3axy$$ The derivative at origin is $0/0$ or indeterminate, found after implicit ...
2
votes
1answer
33 views

Correct algorithm for finding regions of increasing / decreasing and the relationship to critical points.

Yet again, I find myself confused about something that seems basic: critical points and regions of increasing / decreasing. Previously, I thought that to identify regions of increasing / decreasing, ...
2
votes
2answers
47 views

Finding the slope of a line, slightly confused by the answer

(just for context, this is from a study booklet for a military test. I haven't done algebra in about 10 years. I googled around but was having trouble finding specific information about the below.) I ...
2
votes
1answer
33 views

Why can a slope field include points outside the domain of the original function?

I'm looking specifically at the slope field for $y'=\frac{2x}{y}$, which is the derivative of the function $2x^2-y^2=1$ (one of the solutions). But for no "family of functions" is a point, say $(0,1)$ ...
2
votes
2answers
235 views

Slope Intercept Form Word Problem

The speed at which you drive a car can affect the car's fuel economy. The July 2008 Consumer Reports magazine reported the Toyota Camry has a fuel economy of 40 miles per gallon (mpg) at 55 miles per ...
2
votes
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 ...
2
votes
2answers
3k views

How to write an equation of a line bisecting an angle in terms of the slope of the bisector.

I have two lines $y-8=\dfrac{-1}{7}(x+6)$ $y-8=\dfrac{-1}{2}(x+6)$ They both intersect at point $(-6,8)$. I'm trying to find the slope of the line that bisects these two lines. In this question, ...
2
votes
2answers
56 views

How fast is the tree growing?

This information is given: Mrs. Fitzgerald planted a tree. After 10 days, the tree measured 39 inches tall. After 28 days, the tree measured 51 inches tall. This is what I did: Since we are given two ...
2
votes
0answers
65 views

Finding a curve that is orthogonal at $(1,1)$ to the set of given parabolas

I am given a DE like the following: $$\frac{dy}{dx} = \frac{2xy}{x^2-1}$$ When one solves it: $$y = A(x^2-1)$$ Then we obtain an equation for a family of parabolas all intersecting $(-1,0)$ and $(...
1
vote
4answers
674 views

Is this SLOPE of the line even possible?

Is it even possible to build a line having slope of $3$? Could it be a mistake?
1
vote
2answers
1k views

What is the meaning of “slope of the line at a point”

I am new to calculus and until now i knew that slope of a straight line is the rate of change of the y-coordinate with respect to the change in x-coordinate of the straight line or the rise over run ...

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