Short Questions (Question 17 – 19)


Question 17:

In the diagram above, the straight line PR is normal to the curve   y=x22+1 at Q. Find the value of k.

Solution:
y=x22+1dydx=xAt point Q, x-coordinate=2,Gradient of the curve, dydx=2Hence, gradient of normal to the curve, PR=12302k=126=2+kk=8



Question 18:
The normal to the curve y = x2 + 3x at the point P is parallel to the straight line 
y = x + 12. Find the equation of the normal to the curve at the point P.

Solution:
Given normal to the curve at point P is parallel to the straight line y = –x + 12.
Hence, gradient of normal to the curve = –1.
As a result, gradient of tangent to the curve = 1

y = x2 + 3x
dydx = 2x + 3
2x + 3 = 1
2x = –2
x = –1
y = (–1)2+ 3(–1)
y = –2
Point P = (–1, –2).

Equation of the normal to the curve at point P is,
y – (–2) = –1 (x – (–1))
y + 2 = – x – 1
y = – x– 3




Question 19:
Given that   y=34x2 , find the approximate change in x which will cause y to decrease from 48 to 47.7.

Solution:
y=34x2dydx=(2)34x=32xδy=47.748=0.3Approximate change in x to yδxδydxdyδx=dxdy×δyδx=23x×(0.3)δx=23(8)×(0.3)y=4834x2=48x2=64x=8δx=0.025