Differentiation e

CALCULUS I
YILDIZ TECHNICAL UNIVERSITY
FATIH TEMIZ
Differentiation
Recall the definitions of limits and continuity. See the previous application lesson.
1. Determine the continuity of the function f (x) =
3|x − 2|
. Classify the discontinuous points if there
x2 (4 − x2 )
exist.
3x+4
2. Determine whether the function f (x) =
x = 2 or not.
e x−2 has limit at x = 2. Also determine if it is continuous at
Derivative Function
The derivative of the function f (x) with respect to the variable x is the function f 0 whose
value at x is
f (x + h) − f (x)
f (z) − f (x)
lim
= lim
z→x
h→0
h
z−x
provided the limit exists.
3. By using the definition of derivative, find the derivative of the given functions below
p
(a) f (x) = x2 + 1
(b) f (x) = tan(2x) at x = π/6
(c) |f (x)| ≤ x2 at x = 0
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CALCULUS I
YILDIZ TECHNICAL UNIVERSITY
FATIH TEMIZ
(d) at x = 0
(√
x(1 − cos x) x ≥ 0
sin x
x<0
3
f (x) =
4. Find the values of a and b in order to make the function given below differentiable at x = 0.
(
ax + b
x<0
f (x) =
2 sin x + 3 cos x x ≥ 0
5. Find the equation of the tangent to the curve
1
1
1
(a) y sin( ) + x cos( ) = −2x at the point P (0, )
y
y
π
(b) x2 y + y 2 = 3x2 at the point (2, 2)
(c) cos(x − y) = xex at the point P (0,
π
)
2
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CALCULUS I
YILDIZ TECHNICAL UNIVERSITY
FATIH TEMIZ
Linearization
If f is differentiable at x = a, then the approximating function
L(x) = f (a) + f 0 (a)(x − a)
is the linearization of f at a. The approximation
f (x) ≈ L(x)
of f by L is the standard linear approximation of f at a.
6. Find the approximate value of
√
3
28 by using differential calculus or linear approximation.
7. Let g : R → R be a differentiable function with g(2) = −4 and g 0 (x) =
value of g(2.05) by using linear approximation.
√
Derivatives of Trigonometric Functions
d
(sin x) = cos x
dx
d
(tan x) = sec2 x
dx
d
(cot x) = − csc2 x
dx
d
(cos x) = − sin x
dx
d
(sec x) = sec x tan x
dx
d
(csc x) = − csc x cot x
dx
8. Find the derivative of the functions w.r.t. x
(a) (1 + ln x)x
(b)
x3
p
+
3 (1 + x2 )3
q
3
x+
√
x
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x2 + 5. Find the approximate
CALCULUS I
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(c) xy = cot(xy)
(d)
p
3
1 + cos(2x)
(e) x2 y + xy 2 = 6.
9. Let f (x) and g(x) be two differentiable functions. Assume that f (g(x)) = x and f 0 (x) = 1 + (f (x))2 .
1
.
Then show that g 0 (x) =
1 + x2
10. Determine whether the function f (x) = arcsin(
is constant, find it.
√
1−x
) + 2 arctan( x) for x ≥ 0 is constant or not. If it
1+x
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