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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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. ...
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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?
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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 $... 1answer 104 views The slope of$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. $$\mu^n_m(... 1answer 708 views How to find tangents to curves at points with undefined derivatives I will explain my question with the help of an example. We need to find the tangent at origin to the curve$$x^3 + y^3 =3axy$$The derivative at origin is 0/0 or indeterminate, found after implicit ... 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, ... 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 ... 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) ... 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 ... 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 ... 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, ... 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 ... 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$(...
Is it even possible to build a line having slope of $3$? Could it be a mistake?