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Bài tập CALCULUS 67
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Created by T. Madas
Question 231
(****)
y
f ( x) =
x+4
x
M
R
O
1
N
x
The figure above shows the curve C with equation
f ( x) =
x+4
, x>0.
x
a) Determine the coordinates of the minimum point of C , labelled as M .
The point N lies on the x axis so that MN is parallel to the y axis. The finite region
R is bounded by C , the x axis, the straight line segment MN and the straight line
with equation x = 1 .
b) Use the trapezium rule with 4 strips of equal width to estimate the area of R .
c) Use integration to find the exact area of R .
d) Calculate the percentage error in using the trapezium rule to find the area of R .
e) Explain with the aid of a diagram why the trapezium rule overestimates the area
of R .
C2R , M ( 4, 4 ) , area ≈ 12.7344... , area = 38 , 0.53%
3
Created by T. Madas
Created by T. Madas
Question 232
(****)
The curve C with equation y = f ( x ) passes through the point P (16, −5 ) , and its
gradient function f ′ ( x ) is given by
f ′( x ) =
x−6
, x>0.
x
a) Find an equation of the tangent to C at P .
b) Determine an equation of C .
The point Q lies on C and the gradient of C at that point is −1 .
c) Find the coordinates of Q .
3
1
(
C1U , 2 y = 5 x − 90 , y = 2 x 2 − 12 x 2 + 1 , Q 4, − 55
3
3
3
Created by T. Madas
)
Created by T. Madas
Question 233
(****)
y = x2 − 1
y
1
y = 9 1 − 2
x
R
O
x
The figure above shows the graphs of the curves with equations
y = x 2 − 1 and
1
y = 9 1 − 2 .
x
The finite region R is bounded by the two curves in the 1st quadrant, and is shown
shaded in the figure above.
Determine the exact area of R .
C2V , 16
3
Created by T. Madas
