# 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?