In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.
A line is increasing if it goes up from left to right. The slope is positive, i.e.
m
>
0
{\displaystyle m>0}
.
A line is decreasing if it goes down from left to right. The slope is negative, i.e.
m
<
0
{\displaystyle m<0}
.
If a line is horizontal the slope is zero. This is a constant function.
If a line is vertical the slope is undefined (see below).The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances, where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
In mathematical language, the slope m of the line is
m
=
y
2
−
y
1
x
2
−
x
1
.
{\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}.}
The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function
m
=
tan
(
θ
)
{\displaystyle m=\tan(\theta )}
Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.
As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.
This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.
Homework Statement
A curve is given by the equation: y^3+1004=(e^x+1)^2
Find the slope of the tangent line at the point (0,-10).
Homework Equations
The Attempt at a Solution
I took the derivative of ((e^x+1)^2-1004)^(1/3) and that is (2e^x(1+e^x))/(3((1+e^x)^2-1004)^(2/3)) but...
"Prescribed slope"
Ok, so the prof gave us a list of topics and sample questions to focus on for the midterm. This is one of them:
Find all values of x where the graph of a given function has a prescribed slope.
I take this to mean, "find all the values of x where the given function is...
I wasn't sure where to put this, as it is a question pertaining to logic, but not mathematical logic per ce.
I don't understand why the slippery slope argument is considered a fallacy, especially in a political context.
While it is obviously incorrect to say that some action inevitably...
Homework Statement
Find the slope of the tangent line to the curve of intersection of the vertical plane x - y + 1 =0 and the surface z = x2+y2 at the point (1, 2, 5)
Homework Equations
Gradients, Cross products
The Attempt at a Solution
I'm pretty lost here. I think I have to...
Homework Statement
Find the slope of the tangent line to the given polar curve at the point specified by the value of θ.
r = 9sin(θ)
θ = pi/6
Homework Equations
dy/dx = (dy/dθ) / (dx/dθ)
x=rcosθ
y=rsinθ
(sinx)^2 = (1/2)(1-cos2x)
(cosx)^2 = (1/2)(1+cos2x)
2sinxcosx = sin(2x)
The Attempt at...
Homework Statement
f(x) = 2x2 + 4e(5x)
is invertible. Give the slope of the normal line to the graph of f-1 at x = 4.
Homework Equations
(Given in question)
The Attempt at a Solution
I don't know how to solve this question. But , I found the following:-
f(4) = 32 + 4e20
let y...
Homework Statement
Suppose the hoop were a tire. A typical coefficient of static friction between tire rubber and dry pavement is 0.88. If the angle of the slope were variable,
what would be the steepest slope down which the hoop could roll without slipping?
The Attempt at a Solution...
Homework Statement
"for a cantilever beam with a concentrated load in the middle of the span, prove that
the slope is WL2/8EI, and the deflection is 10WL3/96EI"
Homework Equations
the length of the span is L, the concentrated load is acting in the middle at L/2
The Attempt at a...
The surface given by x = x^2 - y^2 is cut by a plane give by y = 3x. producing a curve in the plane. Find the slope of this curve at the ppoint (1,3,-8)
A) 3
B) -16
C) - 8sqrt(2/5)
D) 0
E) 18/sqrt(10)
So we want to look at this curve when y = 3x. Then x = x^2 -y^2 becomes x = x^2 -...
Homework Statement
A skier is going up a slope of 5 degrees to the horizontal. She is skating so only her skis provide propoltion. The static and kenetic friction coefficients for this situation are \mu_{s}=0.12 and \mu_{k}=0.07
Find the magnitude of her maximum possible uphill...
Homework Statement
http://www.screencast.com/users/ntrinh3/folders/Jing/media/7b6a2810-2827-44c3-adfc-cd27d564b812
Homework Equations
Wnet=change in KE
The Attempt at a Solution
so I made the coordinate with x positive in right direction and y is positive up.
Fdcos30-mgh=.5mv^2...
Homework Statement
A 16.0kg box slides 4.0m down the frictionless ramp shown in the figure, then collides with a spring whose spring constant is 240N/m.
Figure Attached
It is a two part question, I got neither and don't understand why.
a)What is the maximum compression of the...
Homework Statement
Starting from rest, a 75 kg skier slides down a 17.0 º slope. If the coefficient of kinetic friction between the skis and snow is
0.120 and it takes 28.2 s to get to the bottom, how long is the ski trail?
Homework Equations
F = ma
s_f = s_i +v_i + 1/2at^2
The...
1. A moving particle has position (x(t),y(t)) at any time t. The position of the particle at t =1 is (2,6), and the velocity vector at any time t > 0 is given by (1 - (1/(t^2)), 2 + (1/(t^2))).
The particle approaches a line as t -> infinity. Find the slope of the line. Show the work that...
1. Prove that the slope of an isochoric process in a T-S diagram is T/Cv where T is the temperature and Cv is the heat capacity at constant volume
2. dQ = T * dS (I understand this)
dQ = Cv * dT for isochoric processes (I don't understand this)
3. Since the two equations share dQ...
Im doing a lab about acceleration due to gravity.
And one of the things we had to do is make a a-t graph using the Aave and Ttotal.
Interval ∆t Ttotal ∆d Dtotal Vave ∆v Aave(cm/s^2)
1 0.050 0.050 2.0 2.0 40 - -
2 0.050 .100 4.0 6.0 80 40 800
3 0.050...
Homework Statement
A large empty crate, with its lid in place, has a 9.50 kg mass hanging from a string attached to the center of the lid. If the crate were sitting (at rest) on a flat surface, the string would simply hang straight down such that the mass would not be touching the floor of...
Homework Statement
Here is the full question.
Consider a curve defined by 2y^3 + 6x^2y - 12x^2 + 6y = 1 and dy/dx = (4x-2xy)/(x^2+y^2+1).
The line through the origin with slope -1 is tangent to the curve at point P. Find the x - coordinate and y - coordinate of point P.
Homework...
Homework Statement
Write the slope intercept forms of the equation of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line.
Point: (2,1) Line: 4x-2y=3
Homework Equations
y-y1=m(x-x1)
y=mx+b
The Attempt at a Solution
I...
Homework Statement
I made a graph of Absorbance vs. Concentration from data I got in the lab.
I graphed it and now am trying to figure out how to find the slope of the curve.
It's supposed to be a Beer's Law calibration curve.
Homework Equations
The Attempt at a Solution
I...
Homework Statement
A skier is pulled up a slope at a constant velocity by a tow bar. The slope is inclined at 30.9 ° with respect to the horizontal. The force applied to the skier by the tow bar is parallel to the slope. The skier's mass is 61.4 kg, and the coefficient of kinetic friction...
Homework Statement
An object is pulled at a constant F. KE0+PE0=0, so W=PE+KE. If there were no friction, the slope of a graph (KE+PE=Y-axis, W=X-axis) would be 1 and the y-intercept would be 0. What would the addition of friction do to the slope and y intercept (would the y intercept be more...
Q1. A cyclist can develop power of 400W and a top speed on a level road of 7.5m/s. The drag force against which the cyclist is working is?
Q2. A constant force drags a block of mass 2kg up a smooth slope at a constant speed of 1m/s, exerting power or 10W. The angle of the slope is?
2...
Homework Statement
A 1.0 kg physics book is on a 20 degree slope. It is connected by a string to a 500 g coffee cup dangling at the bottom side of the incline. The book is given a push up the slope and released with a speed of 3.0 m/s. The coefficients of friction are us = .50 and uk = .20...
Homework Statement
f(x) = 2x2-5x-12 is given;
part a: find derivative of f(x) using first principles,
part b: find the rate of change of f(x) at x=1,
part c: the points at which the line through (1, -15) with slope m cuts the graph of f(x),
part d: the value of m such that the points of...
Why do we take slope=rise/run (or y/x)?
Is it just a definition, or does it have a special significance?
Why can't we take slope as run/rise (i.e. x/y)?
Homework Statement
2. Homework Equations + questions
1) Normal Force
when considering mg as the vertical force,
mg = N cos θ
however, when considering N as the vertical force,
N = mg cos θ
note: θ is the same due to opposite angle
which equation is correct?
2) Acceleration down...
Homework Statement
Find the slope of the tangent line using a specific formula
g(x)=3t-t2
at (0,0)
Homework Equations
Im told to use this equation by the book
f(c+deltax) - f(c)
Deltax
The Attempt at a Solution
Everytime i plug it in by way of the books style i...
Homework Statement
I have this problem on my physics exam and I hoped you guys could help. The Question is: Why is the relationship between the average velocity and the height of a ball rolling down a slope a square root function?
The slope is 100cm long, the height of the ball at the top of...
Homework Statement
A spring at top of a 30 degree slope is compressed 0.5m by a 0.5 kg book. If the book is released, it will just reach the edge of the slope at the spring's equilibrium point, and then it will start to slide down the slope. Assume the spring constant is 25 kg/s^2. Assume...
Homework Statement
It's been a great day of new, frictionless snow. Julie starts at the top of the 60 degree slope shown in the figure . At the bottom, a circular arc carries her through a 90 turn, and she then launches off a 3.0--high ramp. How far horizontally is her touchdown point from...
Homework Statement
I need to find the relationship between object-lens distance and image-lens distance for a lense(converging) but cannot graph the relationship in a way that the focal length is the slope for the graph
Homework Equations
1/u + 1/v = 1/f
where u= object lense distance...
Homework Statement
A 4.7 kg block is placed on an incline with a 14° angle.
Ignoring friction, what is the acceleration of the block?
If the coefficient of kinetic friction is 0.25 (and has overcome static friction) what is the acceleration of the block?
If the incline is 4.6m long, what is the...
Homework Statement
I am given the formula A = Bt / (3d2)
d is what we changed, A is what was measured.
I had to plot ln(A) vs ln(3d2).
What do the slope and intercept of this graph represent?
Homework Equations
The Attempt at a Solution
ln (A) = ln (Bt / 3d2) = ln(Bt) -...
Homework Statement
# 23 http://img26.imageshack.us/img26/1008/questionsh.jpg
Homework Equations
m1 * m2 = -1
The Attempt at a Solution
i get -4 not sure what's wrong:
http://img4.imageshack.us/img4/6092/15608741.jpg
Homework Statement
A small block slides down a slanted board when released. The upper half of the board is smooth and the lower is rough, so that the acceleration of the block on the smooth half is three times greater than it is on the rough half. The block reaches the bottom of the board in...
1. The problem statement, all
variables and given/known data
A clerk moves a box of cans down an
aisle by pulling on a strap attached to
the box. The clerk pulls with a force
of 190.0 N at an angle of 28° with the
horizontal. The box has a mass of 37
kg, and the...
Homework Statement
Find the x values of the points on the curve f(x) = (x^3) + (3x^2) + 3x + 6 where the tangent has a slope equal to m = 6
Homework Equations
The Attempt at a Solution
This question was on an online quiz for my into calculus course... can't seem to wrap my head around...
Homework Statement
y\prime=4x^3+6x^2+2x+1
Homework Equations
The Attempt at a Solution
4x^3+6x^2+2x+1
-1 = 2x(2x^2+3x+1)
(I moved the +1 over so I don't get 1/x, but I figured this should screw up the equation since it is no longer 0 on the left hand side.)
-1 = 2x(2x+1)(x+1)
So the...
http://www.math.rutgers.edu/~greenfie/webstuff/pdfstuff/w6W.pdf
is the problem
For the first part, I believe it is just tracing, right? I look at the line at (0,1) and then follow it, but it's a little confusing.
Next
It says how many critical points...well a critical point is when it is...
Homework Statement
You, a 75-kg skier, glide straight down a snow-covered slope inclined at 15 degrees to the horizontal over spring break. Let's be realistic, what is your acceleration(magnitude and direction) Assume your skis are wood and the snow is dry. mu_k on snow is 0.060.
Homework...
ok guys, i got one that wants me to figure out how much work is done:
A skier of a mass 79.1 kg, starting from rest, slides down a slope at an angle of 38 degrees with the horizontal. The coefficient of kinetic friction, u, is 0.09. What is the net work (the net gain in kinetic energy) done on...
Homework Statement
How should the variables ( l and T) be plotted to obtain k from the slope of a linear graph? Identify (write out) the constants correstponding to the slope and intercept of the linear graph.
Homework Equations
l = lambda
l = (k/f)*(T/u)^0.5
The Attempt at a...
Homework Statement
Ok so the problem is this: A car of a mass 960kg is free-wheeling down an incline (15 degrees to the horizontal) at a constant speed of 9.0 m s^-1
- Deduce that the average resistive force acting on the car is 2.4*10^3N
Homework Equations
F=ma I suppose, but it hasn't...
Homework Statement
I have attached a picture of the problem to this thread, I am having trouble with part c. I am getting an answer which is much larger than 24.5N
The systems shown in the figures are in equilibrium. If the spring scales are
calibrated in Newtons, what do they read? (assume...
This is the only question in my physics section of projectiles I am having trouble with (This is ranked as the hardest question in this section of my text, please cut me some slack :P), because I have no idea how I would go about finding the intersection point of the skier with the slope. The...
Homework Statement
A boy is standing on the peak of a hill (downhill), and throws a rock, at what angle from himself to the horizontal should he throw the rock in order for it to travel the greatest distance.
Answer clues:
1. if, the angle from the slope to the horizontal = 60, then the...