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Bài tập CALCULUS 33
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Created by T. Madas
Question 123
(***+)
3
−1
f ( x ) = x 2 − 2 x 2 + 1 , x > 0
Find the exact value of f ′ ( 4 ) .
f ′ ( 4 ) = 33
8
Question 124
(***+)
The curve C has equation
y=
A
+8 x , x > 0,
x
where A is a non zero constant.
4
The normal to the curve at C , at the point where x = 4 , has gradient − .
3
Find the value of A .
A = 20
Created by T. Madas
Created by T. Madas
Question 125
(***+)
Find the exact value of
2
1
(3 + 2 x )
2
dx ,
giving the answer in the form a + b 2 , where a and b are integers.
7 + 16 2
Question 126
(***+)
A cubic curve C passes through the points P ( −1, −9 ) and Q ( 2,6 ) and its gradient
function is given by
dy
= 3 x 2 + kx + 7 ,
dx
where k is a non zero constant.
Find an equation for C .
C1Z , y = x3 − 5 x 2 + 7 x + 4
Created by T. Madas
Created by T. Madas
Question 127
(***+)
y
C
y = 8 + 4 x − 2 x 2 − x3
A
O
x
B
The figure above shows part of the curve with equation
y = 8 + 4 x − 2 x 2 − x3 .
The curve meets the x axis at A and B .
a) Verify that the coordinates of A are ( −2,0 ) and hence use algebra to show that
the coordinates of B are ( 2,0 ) .
The point C is a stationary point of the curve.
b) Use calculus to determine the exact coordinates of C .
c) Find the exact area of the finite region bounded by the curve and the x axis.
(
)
C2E , C 2 , 256 , area = 64
3 27
3
Created by T. Madas
