Integration Long Questions (Question 1 & 2)


Question 1:
A curve with gradient function 5x 5 x 2  has a turning point at (m, 9).
(a) Find the value of m.
(b) Determine whether the turning point is a maximum or a minimum point.
(c) Find the equation of the curve.

Solution:
(a)
dy dx =5x 5 x 2 At turning point ( m,9 ),  dy dx =0. 5m 5 m 2 =0 5 m 2 =5m m 3 =1 m=1

(b)
dy dx =5x 5 x 2 =5x5 x 2 d 2 y d x 2 =5+ 10 x 3 When x=1,  d 2 y d x 2 =15 (> 0) Thus, ( 1,9 ) is a minimum point.

(c)
y= ( 5x5 x 2 )  dx y= 5 x 2 2 + 5 x +c At turning point ( 1,9 ), x=1 and y=9. 9= 5 ( 1 ) 2 2 + 5 1 +c c= 3 2 Equation of the curve: y= 5 x 2 2 + 5 x + 3 2



Question 2:
A curve has a gradient function kx2 – 7x, where k is a constant. The tangent to the curve at the point (1, 3 ) is parallel to the straight line  y + x – 4 = 0.
Find
(a) the value of k,
(b) the equation of the curve.

Solution:
(a)
y + x – 4 = 0
y = – x + 4
m = –1

f ’(x) = kx² – 7x
Given tangent to the curve at the point (1, 3) parallel to the straight line
k (1)² – 7 (1) = –1
k – 7 = –1
k = 6

(b)
 f'( x )=6 x 2 7x f( x )= ( 6 x 2 7x )  dx f( x )= 6 x 3 3 7 x 2 2 +c 3=2 ( 1 ) 3 7 ( 1 ) 2 2 +c    at point ( 1,3 ) c= 9 2 f( x )=2 x 3 7 x 2 2 + 9 2