Gradient (or slope) of a Line, and Inclination

The gradient (also known as slope) of a line is defined as

gradient=vertical risehorizontal run

In the following triangle, the gradient of the line is given by: ab

In general, for the line joining the points (x₁, y₁) and (x₂, y₂), we have:

We can now write the fomula for the slope of a line.

Slope Formula

Definition

The slope m of the line through the points (x1, y1) and (x 2, y 2) is given by

Example Problem

Find the slope of the line segment joining the points ( 1, - 4 ) and ( - 4, 2 ).

Solution

Label the points as x1 = 1, y1 = - 4, x2 = -4, and y2 = 2. To find the slope m of the line segment joining the points, use the slope formula : So, m = - 6/5.

Straight-Line Equations: Slope-Intercept Form

Straight-line equations, or "linear" equations,graph as straight lines, and have simple variableexpressions with no exponents on them. If you see an equation with only x and y — as opposed to, say x² or sqrt(y) — then you're dealing with a straight-line equation.
There are different types of "standard" formats for straight lines; the particular "standard" format your book refers to may differ from that used in some other books. (There is, ironically, no standard definition of "standard form".) The various "standard" forms are often holdovers from a few centuries ago, when mathematicians couldn't handle very complicated equations, so they tended to obsess about the simple cases. Nowadays, you likely needn't worry too much about the "standard" forms; this lesson will only cover the more-helpful forms.

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I think the most useful form of straight-line equations is the "slope-intercept" form:

y = mx + b

This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept. (For a review of how this equation is used for graphing, look at slope and graphing.)
I like slope-intercept form the best. It is in the form "y=", which makes it easiest to plug into, either for graphing or doing word problems. Just plug in your x-value; the equation is already solved for y. Also, this is the only format you can plug into your (nowadays obligatory) graphing calculator; you have to have a "y=" format to use a graphing utility. But the best part about the slope-intercept form is that you can read off the slope and the intercept right from the equation. This is great for graphing, and can be quite useful for word problems. C

Increasing and Decreasing Functions

Increasing Functions

A function is "increasing" when the y-value increases as the x-value increases, like this:

It is easy to see that y=f(x) tends to go up as it goes along.

Flat ?

What about that flat bit near the start? Is that OK?

• Yes, it is OK when we say the function is Increasing
• But it is not OK if we say the function is Strictly Increasing (no flatness allowed)

Using Algebra

What if we can't plot the graph to see if it is increasing? In that case we need a definition using algebra.

Derivatives of the Sine, Cosine and Tangent Functions

It can be shown from first principles that:

d(sin x)dx=cos x
d(cos x)dx=−sin x
d(tan x)dx=sec2x

In words, we would say:

The derivative of sin x is cos x,
The derivative of cos x is −sin x (note the negative sign!) and
The derivative of tan x is sec²x.

Now, if u = f(x) is a function of x, then by using the chain rule, we have:

d(sin u)dx=cos ududx
d(cos u)dx=−sin ududx
d(tan u)dx=sec2ududx

Example 1

y=sin(x2+3).