0
3.lim x
x
(
1
t
)
t
1 3
dt
x3 x 3 3 3
Fill in the blanks:
1.
x2 et2 t dt
lim 0
x0
x2
=________.
2.
db
dx a f (t)dt
=________.
3.Iff (x) ex,then
f (ln x)dx =________. x
1 e2x
arctan x
x2 (1 x2 )
dx
2 x2 dx
Example Let R be the region bounded by the curve
y ex , and the lines y 0, x 0.
(1) Evaluate the area of the region R
(0, a) . If f (a) 0 , then for every real R
there is at least one numberc in (0, a) for which
Rf (c) cf (c) 0
ab a ab
2.Show that
a
ln. b
b
for 0 b a.
Example If f is continuous,find
Review
一
0 , , , 0 ,1 , 00, 0, 00. 0
b. 0 0
Example
x arcsin x
lim
x0
(arcsin x)3
lim
x0
etan
x x3
e
x
L’Hospital’s Rule
Example
1
ex
lim
x x0
t1 x
lim 1 tan x 1 sin x
x2 3 f (x)
x3
Optimization Problems Example
A manufacturing plant has a capacity of 30 articles per week. Experience has shown that
n articles per week can be sold at a price of
x0
x3
Example
x 12
[t f (u)du]dt
2. lim 12 x12
t
(12 x)3
Where f is differentiable and
lim f (x) 0, lim f (x) 1000.
x12
x12
d.
Example
x t 2et2 dt
lim
x
0
xe x2
c in (a,b) for which
f (c) 0.
4.Show that (x c) x c for x 0,c 0,0 1
5.Show that e x ex for x 0.
Applications of differentiation
Example
Sketch the graph of
p dollars each where p 10 0.15n
and the cost of producing n articles is
30 3n dollars. How many articles should be made
each week to give the largest profit?
Solution
Since
thus
lim ( 3
x
1 x3
1
a
b x
)
0
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therefore 1 a 0 , a 1 ,
lim
1
x 3 (1 x3)2 x 3 1 x3 x2
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The Mean Value Theorem 1. Let f (x) be continuous on [0,a] , differentiable on
(2)Find the volume of the solid generated by revolving the R about the x-axis.
f (a h) 2 f (a) f (a h)
lim
h0
h2
3. Let f (x) be continuous on [a,b], f (x) exists on
(a,b) . If f (a) 0 f (b) , and f (c) 0, c in
(a,b), Show that there is at least one number
x
x
lim ( x x x x)
xห้องสมุดไป่ตู้
f. 0
Example lim( arc tan 2x2)x2
x 2
g. 1
lim(1 1 )x e
x
x
Example
lim(1
tan
x
)
1 x3
x0 1 sin x
lim(
arcsin
x
)
1 x2
x0
x
lim( a1x a2x
anx
)
1 x
Integrals
1.Show that
1
21
7
17
1 1 x4 dx 24
Example P435,5,6,15,10,17
Integrals
1.Find f ( xi)f
f (x) e2x
x
f (t)dt
0
2、Find dy |x0 and y(0) if
x y
x2
y(
t )dt ex
1
1
1
11
lim(
)
dx
n n2 n n2 2n
n2 n2 0 x 1
Integrals
1.The Substitution Rule
1
Example
1 e x dx
1 ex 1 dx 1 e2x
1 x arcsin x(1 x)
x dx
Integrals Example
e x (1 e x ) dx
lim x 3sin x x 3x 2 cos x
e.
Example
lim(ln(1 ex ) x)
x 1
lim(xex x)
x
lim( x2 x x)
x
lim( 1 1 ) x1 ln x x 1
lim(x ln x) lim x ln x
x
x x
Example lim[x x2 ln(1 1)]
x0
n
(a1 a2 an 0)
0 h. 0
Example lim (tan 2x)x x0
i.
0
Example
lim (ln 1)x
x0
x
lim (1 1)x
x0
x
l. Example
lim( 1
1
1
1
)
1
dx
n n2 n n2 2n
n2 n2 0 x 1
Example Find a , b , such that