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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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Can you find an equation parallel or perpendicular to a line when it is not in slope intercept form?

For example, if I need to find the equation of the line parallel to $$2x-3y=4$$ which passes through the point $(1,-5)$ I know how to do this by putting it into slope-intercept form first to find the ...
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Gradient of a velocity vector

Apologies if this is extremely simple. I'm a biology undergrad and I need this to measure cell membranes but I'm really struggling to find a solution anywhere. I'm trying to find the gradient of a ...
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1answer
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The prolate cycloid

A cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid ...
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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 ...
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1answer
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Decrease at an Increasing rate

I need a formula that decreases at an increasing rate. I'm guessing that logarithm is involved in the answer. My criteria: A value of 20 must return a value of 20 (low bound) A value of 80 must return ...
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1answer
17 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, ...
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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 ...
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What is the relation of this line $y = ax +b$ with the points $(x_1, y_1), (x_2, y_2)$

I have a two points with coordinates $(x_1, y_1), (x_2, y_2)$ I am given a line $y = ax +b$ where $a = \dfrac{y_2-y_1}{x_2-x_1}$, $b = \dfrac{y_1x_2 - y_2x_1}{x_2-x_1}$ I wish to know the ...
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Why the value of x2 is reducing?

The slope is given by $ \frac{y1-y2}{x1-x2}$ If the slope is given we can also calculate the value of x2. slope= $ \frac{y1-y2}{x1-x2}$ x1-x2= $\frac{y1-y2}{slope}$ x2=x1-$ \frac{y1-y2}{slope}$ ...
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Why is the gradient of $f(x)= x$ the vector $(1,0,0)$?

Why is the gradient of $f(x)= x$ the vector $(1,0,0)$? This would mean that at every point on the straight line $y = x$, I should go horizontally to experience the maximum change in slope. I was ...
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1answer
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Calculating the regression coefficient of a time-based series

If i have a sample (lets say house price in millions over time) where x=1,2,...,14 samples y-values are shown in the image below, with one sample estimated per month. How do i calculate the estimated ...
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1answer
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(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 ...
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4answers
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The formula of Slope. [closed]

The equation of the slope is m= $\frac{\Delta Y} {\Delta X} $. But why not the other way around? Why not m= $\frac{\Delta X} {\Delta Y} $ ?
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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 ...
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3answers
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How do I find ordered pair, given slope of the tangent line?

The function is $f(x) = x^3 + 9x^2 + 36x + 10$ and the slope given is $9$. I found the derivative and set it equal to $9$, but I ended up with $x = (-9,-33)$ and the answer is $(-3,-44)$. I've asked ...
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What is the slope of tangent line to a rotated ellipse at a specific point?

I've seen discussions about this but with too many details left out. I have an ellipse with the standard parameters: $h, k, a, b$ and a rotation angle. (I can convert all that to the general form ...
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3answers
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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 = ...
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About the slope of a parametric curve at a point. (James Stewart Calculus 8th Edition.)

I am reading "Calculus 8th Edition" by James Stewart. Suppose $f$ and $g$ are differentiable functions and we want to find the tangent line at a point on the parametric curve $x=f(t),y=g(t)$, ...
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4answers
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find equation of tangent for function $f(x)$ parallel to a line [closed]

I dont have a clue how to solve this kind of problem. I dont know how to find out slope of $\ln x$. $f(x) = \ln x$ line $p: 2x-y-3=0$ I have searched for slope of $\ln x$, I know how find out ...
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What is the inclination (grade) of the trail?

In the problem: A straight traail with a uniform inclination leads from a hotel, elevation $5000$ feet, to take a lake in a valley, elevation $600$ feet. The Length of the traill is $4100$ feet. What ...
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1answer
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Why each component of gradient which is slope of the curve in itself while keeping other variables constant gives us slope of curve?

My doubt is suppose we assume a 3D space with 2D surface in it given by some function z = f(x,y). Then each component of the gradient is geometrically the slope of the tangent at f on either x-z or y-...
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1answer
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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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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
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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
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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
101 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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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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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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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
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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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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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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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1answer
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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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Sketch the slope field

I need to sketch the slope field for this ODE $ y'=\frac{y+x-6xe^{\frac{y}{x}-2}}{x} $. The first step is $f(x,y)=\frac{y+x-6xe^{\frac{y}{x}-2}}{x}=c $, where $c=const$. Now I have to draw the curves ...
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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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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
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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
80 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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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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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
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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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Slope Intercept Form of a line - Derivation

Question: 1 I have 2 coordinates i.e. $(0,c)$ and $(x,y)$ Assumption y is less than c. Find slope $(c-y) / (0-x) = m $ Simplify to get $y = mx + c$. But the slope of the line connecting $(0, ...
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1answer
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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
25 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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1answer
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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
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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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Formula for finding the slope of a regression line

I want to calculate the gradient of a regression line and have been looking at various websites and books to find the formula for it. My school textbook writes that this is: $$m={S_{xy}\over(S_x)^2}$$...
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1answer
54 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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vertically −a/b units for every 1 unit of movement horizontally as it is to say that linear equations move vertically −a units…

So I have previously asked a question and the user was very much helpful. But I am still having difficulty with the problem. This is what I have done so far. Can someone please guide me in the right ...