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Bài tập CALCULUS 04
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Created by T. Madas
Question 12
(**)
y = 3x 2 − 6 x −
1
x2
+4, x >0.
Find a fully simplified expression for
y dx .
3
C1M , 4 x + x3 − 4 x 2 + x −1 + C
Question 13
(**)
f ( x ) = − x3 + 9 x 2 − 15 x − 13 , x ∈ » .
a) Find the coordinates of the stationary points of f ( x ) .
b) Determine the nature of each of the two stationary points found in part (a).
c) Hence find the range of values of x for which f ( x ) is decreasing.
C2C , min at (1, −20 ) , max at ( 5,12 ) , x < 1 ∪ x > 5
Created by T. Madas
Created by T. Madas
Question 14
(**)
y
y = x3 − 4 x
R2
O
R1
2
x
8
The figure above shows the cubic curve with equation
y = x3 − 4 x , x ≥ 0 .
The curve meets the x axis at the origin O and at the point where x = 2 .
The finite region R1 is bounded by the curve and the x axis, for 0 ≤ x ≤ 2 .
The region R2 is bounded by the curve and the x axis, for 2 ≤ x ≤ 8 .
Show that the area of R1 is equal to the area of R2 .
C2K , proof
Created by T. Madas
Created by T. Madas
Question 15
(**)
The curve C has equation
y=
a) Find an expression for
6
x2
+
5x
−4, x ≠ 0 .
4
dy
.
dx
b) Determine an equation of the normal to the curve at the point where x = 2 .
dy 5 12
= −
, y = 4x − 8
dx 4 x3
Question 16
(**)
f ( x) = 6x + 9 x −
4
x2
, x >0.
Find a fully simplified expression for
f ( x ) dx .
3
C1J , 3x 2 + 6 x 2 + 4 x −1 + C
Created by T. Madas
