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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33 views

Understanding calculating the intercept C between two points

I currently started with some basic geometry and I'm already stuck at some very very basic intuition regarding finding the line between two points in a plane. I understand $y = mx + c$ and I am able ...
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1answer
20 views

How to tell the run of a line segment where the area underneath it, starting point, and slope are known.

I have a program where I am currently solving for the difference of X₁ and X₂ of a line-segment where the area under the line-segment (A), the slope (s), and the starting point (X₁,Y₁) are known. ...
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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 ...
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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(...
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103 views

Use Limits to calculate slope of the tangent

Use limits to calculate the slope of the tangent to the curve $y=\frac1x$ at $x=a$. I need to write an equation for the tangent to $y=\frac1x$ at $x=4$. I think I understand the basics of the ...
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226 views

Finding the slope of the line normal to the graph at a given point

My Problem: What is the slope of the line normal to the graph of $f(x) = e^x-x^e-e$ at the point where the graph crosses the $x$-axis? a. $-0.288$ b. $-0.110$ c. $3.471$ d. $9.106$ I ...
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3answers
70 views

Given only the slope, find the shortest distance from a line to a point on the x-axis

I have the following amusing question. Suppose that the red dot is the point $x = -1$ The blue line is the line crossing the origin with slope $-1/k, k > 0$ I wish to find the length of the ...
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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: $$f(x) = ax^2 + bx +c$$...
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65 views

slope of a linear function in semi-log plot

I have a decreasing linear function. So,the slope of this function df/dt <0. Now, if we plot this function in a semi-log plot with log(t) in horizontal axis and y in vertical axis, can we say that ...
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123 views

How can I find the slope of a line tangent to a small circle on a sphere? [closed]

I draw a circle on the earth, so that it passes through the north pole. I then begin walking around the circle, keeping track of my latitude. How do I tell what direction that I'm facing by knowing ...
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197 views

how to find slope of discrete point?

I am wondering if it is possible to find the slope at each point in the following dataset, ...
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1answer
127 views

How to shorten a line but maintain its angle? [closed]

I have a line that looks like this: to draw this line i have the following: point A: (x,y) point B: (x,y) so for example, i will be given this path: (90,20),(400,300) and then i can draw a line ...
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74 views

What is the unit of this “Rate of Change”?

This is the plot of the function $f(x) = 0.9x$. So, the rate of change of $y$ with respect to $x$ is $0.9$. My question is, what is the unit of this Rate of Change? $0.9$ per $x$?
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1answer
182 views

calculate angle of line with negative slope

I want to use the formula $$ tan(\alpha)=m $$ for negative slopes but always get negative degrees. For instance, say the slope of a line $g$ is $-1$. Using the formula above (arctan$(-1)=\alpha$), I ...
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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 ...
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152 views

Find the value of Δx for which the value of Δy/Δx is within 0.1 units of slope.

Here is the link for the problem that I am trying to solve: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/...
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54 views

When the slope of the angle bisector is 1, is the product of the slopes of the 2 lines forming the angle equal to 1

When the slope of the angle bisector is 1, I have been told that the product of the slopes of the 2 lines forming the original angle is 1. Is this true?
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1answer
25 views

Slope of a triangle where the sum of the height at each integer equals 1.

I have a problem where I'm trying to determine a proof to get the slope of a triangle $m$ in terms of $n$ where $\sum_{i=1}^n h_i = 1$. I think I stumbled apon the answer being $m = \frac{2}{n(n+1)}$ ...
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1answer
17 views

Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not.

Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not. Explain what this plane looks like. Would it be a plane in the xy plane? Not sure how to start this ...
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2answers
31 views

How can I prove if l1 is as twice as steep as l2?

How do I find the slope of each line using SohCahToa? I plan on using the unit circle but I am having difficulty calculating slope.
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73 views

Would parallel lines in 2 parallel planes have the same slope?

If I am given 2 points in Plane A and 2 points in Plane B and I connected those points, would the lines have the same slopes? How can I show that algebraically?
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1answer
72 views

Kink versus jump

Suppose I have a function defined as follows $n(r) = \begin{cases} n_h \quad \text{if} \quad r< \hat{r} \\ n_l \quad \text{if} \quad r \ge \hat{r}\end{cases}$. Where $n_h > n_l$ and $r \in [0,\...
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66 views

Why is the slope of a linear equation of the form $ Ax + By + C = 0 $, $ - \frac {A}{B} $?

The slope of a linear equation of the form $ Ax + By + C = 0 $ (general form) is given by $- \frac {A}{B} $. Could someone intuitively explain why we use this formula? I've read that this is ...
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113 views

Explain why the slope of the equation is defined only for nonzero values of $b$. What happens when $b$ is zero?

Consider the equation $ax+by=c$ in 2-space and the slope for that equation where $a$ and $b$ are real numbers. Explain why the slope of the equation is defined only for nonzero values of $b$. What ...
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2answers
46 views

Consider the equation $ax + by = c$ in 2-space and the slope for that equation where $a$ and $b$ are real numbers.

Consider the equation $ax + by = c$ in 2-space and the slope for that equation where a and b are real numbers. Explain using multiple representations why it is equivalent to say that a linear ...
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749 views

Finding value(s) for x where the tangent line of f(x) has a slope of 4.

I have this equation $\frac{3}{4}x^4 - 4x^3 + \frac{1}{2}x^2 + 7$ I am trying to find the value(s) for $x$ where the tangent line of $f(x)$ has a slope of 4. So I take my first derivative which I ...
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2answers
27 views

How do significant figures work in a cross unit division?

In one of my problems, I needed to find a slope on a graph with different units in physics. 309 cm² - 0.0 cm² / 100 cm - 0.0 cm The answer is A = 3.09 r² with A being area and r² being radius ...
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1answer
146 views

A line in the xy-plane contains the points (5, 4) and (2, –1)

Question: "A line in the $xy$-plane contains the points $(5, 4)$ and $(2, –1)$. Which is bigger: a) the slope of the line or b) $0$." Result: They draw out the figure and say "you can see that the ...
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29 views

How to measure the rate of decrease or increase of a particular stock in a specific time period

I need to measure the drop or increase of the value of a stock within a specific time frame. For example given a company XYZ, I need a formula to calculate the percentage change in value for stock XYZ ...
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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),\ \...
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2answers
220 views

How to find point in Descartes Folium with slope of -1/3?

I stumbled upon a problem in my calculus book that asked to find the point in $x^3 + y^3 = 3xy$ that had a slope perpendicualr to $y = 3x + 1$ and also was in the first quadrant. I began by getting ...
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1answer
196 views

Find the slope of the line using least squares method

This question is taken from GATE 2005 ICE subject paper. Using the given data points given below, a straight line passing through the origin is fitted using least squares method. $$(x, y)$$ ...
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2answers
634 views

Derivative of $\arcsin(\cos(x))$ at $x=0$.

We have to differentiate this function at $x=0$ $$\left. \dfrac{d \arcsin (\cos (x))}{dx} \right|_{x=0}$$ Using the identity $\cos x=\sin \Big(\dfrac{\pi}{2}-x\Big)$ we get $$\left. \dfrac{d \arcsin (\...
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3answers
306 views

Graphing Inequality: 5x+10y >0 Saw the graph answer but I can't understand it.

The answer is a diagonal line that cuts through 0,0 Like in this website: https://www.symbolab.com/solver/step-by-step/5x%2B10y%3E0 My attempt: 5x + 10y(0) >0 5x>0 x>0 5x(0)+10y>0 10y>0 y>0 ...
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0answers
82 views

Justifying a Fundamental Theorm of Calculus Video

In this video, speed, defined as the distance traveled in one unit of time, is represented by rectangles on the distance-time (distance as a function of time) graph. At first, the examples stick to a ...
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41 views

How to see intuitively that for large values of $x$, the tangent from 1st quad. to 3rd quad. on cubic curve makes $270^\circ$.

This is in continuation to my earlier post at : Tangent to Cubic curve has positive slope. It is discussed in earlier post (& before that) that given a cubic curve $y = x^3$, if tangents are ...
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1answer
140 views

Tangent to Cubic curve has positive slope.

This is in continuation to my earlier post at : How many times tangent to a cubic curve $y = x^3$ from a point on it, meets again at another point. A brief summary of the previous post : If take a ...
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1answer
64 views

Find the equation of tangent.

Original Question: Find the equation of the tangent at point P of parabola $x^2=4y$, which is at the minimum distance to center of a circle $x^2+y^2+6x+8=0$. What i did: Found coordinates of center ...
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537 views

Averaging the slope of multiple lines

I have 25 different trend lines. I'm interested in rate of change, which will be given by the slope of these lines. While searching on the internet, i came across this interesting post, What good can ...
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2answers
58 views

how to compare slope of two lines and find the angles in 3d

I have three points $A(x_1,y_1,z_1)$ ,$ B(x_2,y_2,z_2)$ and $C(x_3,y_3,z_3)$ and I want to compare the slopes of AB and BC and check the angles between them. What should I do ? Thanks in advance .
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4answers
627 views

Finding the equation of the line that is orthogonal to a tangent line to a Parabola y = x²

There is a parabola $y = x^2$. The question ask for the "a" of the equation $y = ax +$$3 \over 2$, which is the equation for the line that goes through the point $(0,$$3 \over 2$$)$ and is orthogonal ...
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1answer
571 views

While working with negative slopes, how to decide which slope is greater?

For an example, we take two lines L1 and L2. L1 makes 130 degrees with +ve X axis and has slope m1=tan(130 degrees)= -1.17. Similarly, L2 makes 150 degrees with +ve X axis and has slope m2= -0.577. ...
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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 $(0,1)$ ...
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3answers
191 views

What is intuition behind direction of derivative of a function?

I can't quite grasp the concept of the why gradient of a function points in the direction of steepest ascent. Thinking about it lead me to the basic notions of derivative, but here example first: $$f(...
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52 views

Why slope of line defined as height to length? [closed]

I want to understand: why slope defined as ΔY/ΔX but not ΔX/ΔY ?
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30 views

Find value of $t$ where slope of parametrically-defined curve $=4$ using multivariable calculus

A recent problem I encountered gave me curve $C$ defined by the parametric equations $x(t)=2t^{2}+t-1$ and $y(t)=t^{2}-3t+1$, and asked me to find point $t$ where the slope of the tangent line to $C$ ...
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28 views

Arranging points

How to arrange $n$ points to make sure that they determine an odd number of slopes? I was thinking to break this into the case where $n$ is even and odd. If $n$ is even then we put a line through $n-...
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69 views

How do I calculate the slope of a line knowing one intersection point and the intersection point of that line after two reflection angles?

When a point starting at $O(0,0)$ intersects a side of the square $ABCO$, it reflects so the original angle and reflecting angle are equal. In the bottom side of the square ($AO$), we know the point ...
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2answers
102 views

Slope of the tangent

I have the following exercise. $1$. Find the slope of the tangent to the curve $y = x^3 -4x + 1$ at the point where $x = a$. $2$. Find the equations of the tangent lines in the points $(1, -2)$ ...
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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 ...