limit & derivative problems by anurag tyagi classes (atc)
DESCRIPTION
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achievesuccess in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.MEET US AT:www.anuragtyagiclasses.comTRANSCRIPT
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Limit & Derivative Problems
Problem… Answer and Work…
1.
1. 2
2 4
3 7 11lim
2 4 8x
x x
x x x
1
2
4 4 4
2 4
4 4 4
3 7 110
lim 082 4 8x
x x
x x xx x x
x x x
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Limit & Derivative Problems
Problem… Answer and Work…
2.
2.
16
16lim
4x
x
x
2
16 16
4 4lim lim 4 16 4 8
4x x
x xx
x
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Limit & Derivative Problems
Problem… Answer and Work…
3.
3. 3
2
8lim
2x
xx
3
2
2
2 2
2
2 2 4lim
2lim 2 4 2 2 2 4
4 4 4
12
x
x
x x x
xx x
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Limit & Derivative Problems
Problem… Answer and Work…
4. Consider the function given by
Is f(x) continuous at x=1? Justify.
4.
2 3, x 1( )
3x, x > 1
xf x
4
12
1
1
Does lim ( ) (1)?
lim ( ) 1 3 4
lim ( ) 3 1 3
lim not equal to each other,
theref ore,
not continuous at x = 1
x
x
x
f x f
f x
f x
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Limit & Derivative Problems
Problem… Answer and Work…
5. Is the function given by
continuous for all x? If not, where are the discontinuities? Are they removable?
5.
2, x 3( )
6 - x, x 3
xh x
5
3
3
3
3 3
lim ( ) ( )?
lim ( ) 3 2 5
lim ( ) 6 3 3
not continuous at x = 3
lim ( ) lim ( )
shows that the discontinuity
is not removable
x
x
x
x x
h x h x
h x
h x
h x h x
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Limit & Derivative Problems
Problem… Answer and Work…6. Let the piecewise function f be
defined as follows:
Which of the following is true about the function f?
I. f(2) = 2
II.
III. f(x) is continuous at x = 2
A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III
6. Test: f(2) = 2? Yes, so I is true
Test:
Test: f(x) is continuous at x = 2?
Does the lim f(x) = f(2)?
4 is not equal to 2
No, so III is false
Answer is A) I only
2 4, f or x 2
( ) 22, f or x = 2
xf x x
6
2lim ( ) 2x
f x
2
22
22
lim ( ) 2?
( 2)( 2)lim ( ) lim
( 2)lim ( ) lim 2 4
No, I I is f alse
x
xx
xx
f x
x xf x
xf x x
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Limit & Derivative Problems
Problem… Answer and Work…
7.
What is the value of a for which f(x) is continuous for all values of x?
A. -2
B. -1
C. 0
D. ½
E. 1
7. To be continuous at x = 1
7
2
1, x 1I f ( )
3 , x > 1
xf x
ax
1 12
1 12
2
lim ( ) lim ( )
lim 1 lim3
2 = 3 + ax
2 = 3 + a(1)
2 = 3 + a
-1 = a
x x
x x
f x f x
x ax
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Limit & Derivative Problems
Problem… Answer and Work…
8. Find the cartesian coordinates of the point on the graph of
where the instantaneous rate of change of f is equal to 5
8.
to find y substitute x = ½ in the original function f(x)
Ans: (1/2, 11/4)
8
2( ) 3 2 1f x x x
2( ) 3 2 1f x x x
' ( ) 6 2
5 6 2
3 6
12
f x x
x
x
x
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Limit & Derivative Problems
Problem… Answer and Work…
9. Which of the following directly describes the discontinuities associated with
a. A hole at x = 3, a vertical asymptote at x = 3
b. Holes at x = -3 and x = 3
c. A hole at x = 3, a vertical asymptote at x = -3
d. Vertical asymptotes at x = 3 and x = -3
e. No discontinuities
9.
Hole at x = 3 because we factored out (x – 3)
There is a vertical asymptote at x = -3
9
2
2
2 3( )
9
x xf x
x
( 3)( 1) 1( 3)( 3) 3x x xx x x
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Limit & Derivative Problems
Problem… Answer and Work…10. Given the piecewise function
For what values of a and b is f(x) differentiable at x = 1?
A. a = 2 b = -3
B. a = 2 b = -2
C. a = -2 b = 1
D. a = 3 b = -1
E. a = 5 b = 8
10. Differentiability implies continuity
To be differentiable x = 1
Solve for a when b = 1
a – 1 = -3 a = -2 Ans: C
10
2
2 x 1( )
bx 1 x > 1
x af x
2
2
2 1 when x = 1
2(1) + a = b(1) 1
2 1
3
x a bx
a b
a b
2(2 ) ( 1)
2 2 when x = 1
2 2 (1)
1
d dx a bx
dx dxbx
b
b
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Limit & Derivative Problems
Problem… Answer and Work…
11. Which of the following is (are) true about the function
I. It is continuous at x = 0II. It is differentiable at x = 0III.
A. I onlyB. II onlyC. I and III onlyD. II and III onlyE. I, II, III
11. Test 1: Continuous at x = 0
yes
Test 2: Differentiable at x = 0?
No
Test 3:
Yes
Ans: C
11
13( ) ?f x x
0lim ( ) 0x
f x
3( )f x x
2 3
2 3
1'( )
31
03
undefi ned at x = 0
f x x
x
0
0 0
lim ( ) 0?
lim ( ) 0 lim ( ) 0x
x x
f x
f x f x
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Limit & Derivative Problems
Problem… Answer and Work…12. To apply either the Mean Value
Theorem or Rolle’s Theorem to a function f, certain requirements regarding the continuity and differentiability of the function must be met. Which of the following states the requirements correctly?
A. f is continuous on (a, b) and differentiable on (a, b)
B. f is continuous on (a, b) and differentiable on [a, b]
C. f is continuous on (a, b) and differentiable on [a, b)
D. f is continuous on [a, b] and differentiable on (a, b)
E. f is continuous on [a, b] and differentiable on [a, b]
12. Look at the definition of Rolle’s Theorem and the Mean Value Theorem
f is continuous on [a, b] and differentiable on (a, b)
Ans: D
12
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
A. Determine the x and y intercepts, if any. Justify your answer.
13. A
13
2
1 1( ) 1f x
x x
2
2
2
2
2
2
2
1( )
10
0 1 no real solutions
n
x-intercept
y-intercep
o x-intercepts
0
t
0 1 undefi ned
0no y-intercepts
x xf x
x
x x
xx x
y
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
B. Write an equation for each vertical and each horizontal asymptote. Justify your answer.
13. B
Vertical asymptote
Horizontal asymptote
14
2
1 1( ) 1f x
x x
2 0
0
x
x
2
2 2 2 2
2 20 0
2
11
lim lim 1
1
x x
x xx x x x x
x x
xy
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
C. Determine the intervals on which f is increasing or decreasing. Justify your answer.
13. C
15
2
1 1( ) 1f x
x x
1 2
2 3
3
3
( ) 1
2'( ) 2
0 ( 2)
0 x = -2
Do a sign graph f or the critical points 0, -2
0 is a vertical asympt
Decreasi
ote
ng (- , -2) and (0, )
I ncreasing (-2, 0)
f x x x
xf x x x
x
x x
x
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
D. Determine the relative minimum and maximum points, if any. Justify your answer.
13. D
Relative minimum occurs at x = -2
when x = -2
16
2
1 1( ) 1f x
x x
1 2( 2) 1 ( 2) ( 2)
1 1 31
2 4 43
( 2, )4
no maximum because at x = 0 which
is the vertical asymptote
f
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
E. Determine the intervals on which f is concave up or concave down. Justify your answer.
13. E
17
2
1 1( ) 1f x
x x 2 3
3 4
4
' ( ) 2
"( ) 2 6
2 ( 3)
undefi ned at x=0 because
it is a vertical asymptote
0 = x+3
3
Do a sign graph using critical
Concaves d
points -3,
own (- , -3)
Concaves up (-3,0) (0, )
0
f x x x
f x x x
x x
x
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Limit & Derivative Problems
Problem… Answer and Work…13. Let f be the function defined by
F. Determine any points of inflection
13. F
Point of inflection when x = 3
18
2
1 1( ) 1f x
x x
1 2
2
( 3) 1 ( 3) ( 3)
1 11
7( 3, )
9
3 ( 3)9 3 19 9 979
f
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Limit & Derivative Problems
Problem… Answer and Work…14. On the interval [1, 3], what is the
average rate of change for the functions, if
14.
19
2( ) 3 4 ?s t t t
2
2
(3) (1)3 1
(3) 3(3) 4(3) 27 12 15
(1) 3(1) 4(1) 3 4 1
15 ( 1)8
3 1
s sAvg
s
s
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Limit & Derivative Problems
Problem… Answer and Work…15. Is the function defined by
continuous at x = 4? Justify your answer.
15.
20
3, 3 x < 7( )
5, 7
xf x
x x
4
4
lim 3 4 3 1
(4) 4 3 1
lim ( ) (4)
x
x
x
f
f x f