We now know that the slope of the tangent line is
$-\frac{1}{4}.$ To find the equation of the tangent line, we also need a point on the line. We know that
$f\left(2\right)=\frac{1}{2}.$ Since the tangent line passes through the point
$(2,\frac{1}{2})$ we can use the point-slope equation of a line to find the equation of the tangent line. Thus the tangent line has the equation
$y=-\frac{1}{4}x+1.$ The graphs of
$f\left(x\right)=\frac{1}{x}$ and
$y=-\frac{1}{4}x+1$ are shown in
[link] .
The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the
derivative . The process of finding a derivative is called
differentiation .
Definition
Let
$f(x)$ be a function defined in an open interval containing
$a.$ The derivative of the function
$f(x)$ at
$a,$ denoted by
${f}^{\prime}\left(a\right),$ is defined by
Now that we can evaluate a derivative, we can use it in velocity applications. Recall that if
$s\left(t\right)$ is the position of an object moving along a coordinate axis, the
average velocity of the object over a time interval
$\left[a,t\right]$ if
$t>a$ or
$\left[t,a\right]$ if
$t<a$ is given by the difference quotient
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |.
The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject.
Well, this is what I guess so.