Graphical interpretation of First D...
Graphical interpretation of First Derivative

Tangent Graph

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Tangent graph is not like a sine and cosine curve. This graph looks like discontinue curve because for certain values tangent is not defined.
tan x = sin x / cos x
For some values of x, cos x has value 0. For example, x = π/2 and x = 3π/2. When this happens, we have 0 in the denominator of the fraction and this means it is undefined. So there will be a "gap" in the function at that point. This gap is called a discontinuity.
y = tan x.
we know that tan x is a periodic function with period π This means it repeats itself after each π as we go left to right on the graph. As x increases from 0 to π/ 2, tan x keeps on increasing from 0 to ∞. As x crosses the value π / 2, tan x becomes negative and is arbitrarily large in magnitude. It increases to 0 as x increases from π / 2 to π.
The graph of y = tan x is

Note that there are vertical asymptotes (the blue dotted lines) where the denominator of tan x has value zero.
As there is a phase shift in the sine and cosine graph, in the same way there is a phase shift in tangent graph. y = a tan (bx + c) bx + c = 0 ⇒ x = -c/b that is the first cycle. Period = π/|b| For every cycle add k(π/|b|) that gives you the asymptotes. Example: y = 4tan(2x + π/3) 2x + π/3 = 0 ⇒ x =-π/6 Period = π/b = π/2
Phase shift = -π/6 = -4π/24
Add and subtract π/6
It will be -π/24 and -7π/24

Examples :
1) Find the period of y = tan (3x)
Solution :
The general equation of tangent function is
y = a tan(bx)
a = 1 Period = π/b
Period = π/3
2) y = 3 tan(2x)
Solution :
The general equation of tangent function is
y = a tan(bx)
a = 3 and b = 2
Period = π/b
∴ Period = π/2

1) Draw the graph of y = tan 2x.
2) From the given graph write the function of it.

Tangent graph

Graph Dictionary

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