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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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
26 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
28 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
280 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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1answer
39 views

To check increasing and decreasing function

Can anyone help me in finding the interval for increasing and decreasing of $g(x)$. I know that $tanx$ is increasing in the interval given and $cotx$ is decreasing.I even tried differentiating $g(x)$ ...
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1answer
37 views

Pondering $f(x) = \alpha x$ for an irrational $\alpha$.

So I was helping my little brother with linear equations, specifically the slope of a line, and I stumbled on something that is a little hard for me to wrap my head around. Obviously, we think of ...
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2answers
41 views

Conceptual question on graphs

I'm a student of economics but I have encountered a mathematical doubt while studying some material (nothing new!). I will try to present the question without any reference to economic concepts: I ...
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23 views

On the direction of steepest ascent of a plane in $\mathbb{R}^3$.

Suppose we have a plane in $\mathbb{R}^3$ whose equation is given by $$z=ax+by,$$ and let $\textbf{n}(-a,-b,1)$ be a vector normal to this plane. How can we intuitively explain why the projection ...
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63 views

Prove that only those points satisfying y-y1=m(x-x1) can lie on the line L passing through P1(x1,y1) with slope m.

First question, go easy on me. =) From text: Show that only those points satisfying $y-y_1=m(x-x_1)$ can lie on the line $L$ passing through $P_1(x_1,y_1)$ with slope $m$. I can evaluate $m$ from ...
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2answers
28 views

Why is the slope form written how it is?

So, I was recently reviewing slope. While doing some problems with the slope formula $m = {y_2 - y_1\over x_2 - x_1}$, I started thinking why it was that and not $m = {y_1 - y_2\over x_1 - x_2}$. ...
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1answer
25 views

Slope of line if Δx is equal to 0

To calculate slope of line, I use $m = \frac{Δy}{Δx}$, but it doesn't work if $Δx = 0$. Is there a way to calculate slope of straight line? Is there anything I didn't notice?
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2answers
970 views

Find the equations of all tangent lines to the curve $x^2 + 4y^2 = 8$ that pass through the point (−4, 0). [duplicate]

So I stared off by implicitly differentiating $x^2 + 4y^2 = 8$, and I got $\frac{dy}{dx }= \frac{-x}{4y}$. I then plugged this into the point-slope formula$ (y - y1 = m (x - x1))$ along with my ...
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1answer
24 views

Vehicle weight on a slope?

A vehicle needs to winch in a cable with 1000lbs of pressure/weight. The vehicle weighs 5400lbs How much of a slope must the vehicle park on to achieve 1000lbs of weight to winch in the cable?
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1answer
26 views

Partial Differential Equations - Characteristic Curves

I am going through a book on partial differential equations, and am slightly confused by this paragraph: Consider the equation $\frac{dy}{dx}=p(x, y)$. a) Think of solutions to this equations ...
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1answer
40 views

How to find X value so that the slope of the function is -1 (exponents)?

I was watching 3Blue1Brown playlist of derivatives and integrals, after learning and understanding the visualization of those topics, I wanted to play around with the math. Introduction to the ...
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1answer
43 views

Alternative interpretation of a basic slope word problem?

I ran into this question today: At a restaurant, n cups of tea are made by adding t tea bags to hot water. If t = n + 2, how many additional tea bags are needed to make each additional cup of tea? ...
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3answers
42 views

Unclear with slope field and initial values.

So I am working with a first order ODE: $$\frac{dy}{dx}= y\cos^2(x)$$ We are told to draw by hand three separate solution curves for each initial value points: $y(0)=0, y(0)=\frac{1}{2}, y(0)=-\frac{...
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1answer
54 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, ...
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0answers
21 views

slope equation calculation for a line [duplicate]

when slope m=0 it is a straight horizontal line and when slope is vertical line..I mean what is the value for slope m to get vertical line?I think that for vertical line, slope is m=0 also.
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1answer
15 views

Total Cost According to Miles Driven [closed]

A computer repair person charges 50.00 dollars per hour, plus an additional mileage fee. The charge for mileage varies directly with the square root of the number of miles traveled. If one hour plus ...
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0answers
38 views

Using slopes to find orthocentre of a triangle

Let vertices of a triangle be $A(1,0)$, $B(-2,1)$, $C(5,2)$. Let orthocentre be $H(a,b)$. Slope of $AH=b/(a-1)$, slope of $BH = (b-1)/(a+2)$, slope of $CH = (b-2)/(a-5)$. I assumed $AH, BH, CH$ to be ...
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1answer
62 views

Why did this method gave the orthocentre of the triangle?

Coordinates of vertices of a triangle are given(say $A$, $B$, $C$). I assumed coordinates of orthocenter to be $H(a, b)$. I calculated slope of $AH, BH, CH$ in terms of $a$ and $b$. For some reason I ...
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1answer
161 views

Average Rate of Change From Graph

I'm doing my pre-calculus homework and I cannot find an answer to this problem in my book or anywhere. I am give the following graph The question is the following: Estimate the average rate of ...
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1answer
312 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 ...
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2answers
41 views

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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1answer
155 views

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
224 views

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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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 ...
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1answer
119 views

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
32 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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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 ...
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2answers
46 views

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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0answers
30 views

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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1answer
25 views

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
22 views

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
431 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 ...
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4answers
39 views

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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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 ...
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3answers
330 views

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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3answers
218 views

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
183 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 = ...
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2answers
45 views

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
56 views

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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1answer
78 views

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
22 views

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
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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2answers
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 ...