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t angent
At x = 3 the equation of tangent to y = x3 is : z3(x) = 27x − 54
t angent
At x = 4 the equation of tangent to y = x 3 is : z4(x) = 48x − 128
t angent
Excercise 1 : Draw the graph of y = x 3 and complete the following table. tanθi y’(xi) y(xi) 3.078 3 3 3x − 2 o 12 8 12x − 16 87 o 27 27 27x − 54 88 o 48 64 48x − 128 xi approx. θi 1 72 o 2 85 3
4 101 zi (x) = y ’(xi ) . x + c i /
Exercise 2: Repeat exercise 1 with y(x) = 1 2 x2. What if tan θ = 0 ? , that is to say the tangent to the curve is parallel to the
xaxis. We shall deal with this in the next few chapters. Exercise 3: In our main example of the projectile depicted below with u = 2√2
and θ = π/4 draw the graph and complete the table as in exercise 1 for
instants t1 = T/ 4 , t2 = T/ 2 , and t3 = 3T/ 4 .
height y(t) = u . s i n θ . t  1 g t 2
2
z 3 (t 3 ) = y(t 3 ) ( θ3 (0,0) T/ 2 t3 T t y’(t 3 ) = tan( θ3 )
y ’(t) = u . s i n θ  g t tangent
It does not make sense to talk about a tangent to a WELLBEHAVED linear function.
tang
However, its SLOPE = FIRST DERIVATE = tan θ, when θ is the angle of inter section
angle intersection
ang
the linear function makes with the xaxis.
With the concept of the tangent in mind, the reader may wish to review the bouncing
tangent
bouncing
Analytical geometr
ball e xample on page 91 and gain an Anal ytical g eometr y v iew of
A nal
diff erentia bility.
entiability
dif f er entia bility.
When a SINGLEVALUED and CONTINUOUS function is NOT differentiable at a
differentia
dif entiab
particular point or instant, it may have more than one tangent at that instant. In
example 1 on page 87, f(x) is SINGLEVALUED and CONTINUOUS. But f(x) is NOT
differentiable at x = a. At the coordinates ( a, f(a) ) we may draw infinitely many
tangents.
102 We may relate the Analysis view of differentiable and the Analytical geometr y
Analysis
differentia
Analytical geometr
eometry
Anal
dif entiab
Anal
property of the tangent and say :
tangent
tang
f(x) is DIFFERENTIABLE at a ⇔ f(x) has a UNIQUE TANGENT at f(a). Suppose we know the tangents z1(x), z2(x), z3(x), . . . at x1, x2, x3, . . . respectively,
can we find the curve...
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 Fall '09
 TAMERDOğAN

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