Easy Geometrical Applications of Calculus questions (1 Viewer)

[bawd]

For the following few questions, it asks to sketch, showing any stationary points, inflexion or asymptotes. Can somebody simply show the working out required to find the said points (no sketch needed) and explain how to obtain them them? I am extremely confused when it comes to finding asymptotes, particularly horizontal ones. I will be so grateful for the help. Since there are ten questions, maybe a few BoSers can do one or two questions each?

NOTE: These questions are from Maths in Focus Book 2 HSC Course Revised, Exercise 2.8. I've done some of the questions mechanically and gotten them correct, but without actually understanding why, if you get me.

1. y = 2 / (x^2 +3)

2. y = 1 / (x^2 - 16)

3. y = x^2 / (2x + 3)

4. y = x^2 / (x^2 + 4)

5. y = x^2 / (x^2 - 9)

6. y = 1 + x / (x^2 - 1)

7. y = x / (x^2 + 1)

8. y = (1 +x^2) / (1 - x^2)

9. y = (1 + x^2) / (1 - x^2)

10. y = (2x +3) / (x^2 - 4)

Thanks again!

 

[Timothy.Siu]

asymptotes occur when the denominator equals 0 as this is undefined and also the horizontal one can be found when x--->infinity

to find stationary points, u differentiate the equation and find when f'(x)=0 after this you have to determine the nature of the stationary point so you can either sub it in f''(x) and if it is >0 then it is a local minimum and if it is <0 it is a local maximum and if f''(x)=0 and f'(x)=0 then it is a stationary point of inflexion.

The other way to determine the nature of the stationary point is to test both sides of the stationary point using f'(x) and drawing a table of values.

 

[shaon0]

For the following few questions, it asks to sketch, showing any stationary points, inflexion or asymptotes. Can somebody simply show the working out required to find the said points (no sketch needed) and explain how to obtain them them? I am extremely confused when it comes to finding asymptotes, particularly horizontal ones. I will be so grateful for the help. Since there are ten questions, maybe a few BoSers can do one or two questions each?

NOTE: These questions are from Maths in Focus Book 2 HSC Course Revised, Exercise 2.8. I've done some of the questions mechanically and gotten them correct, but without actually understanding why, if you get me.

1. y = 2 / (x^2 +3)

2. y = 1 / (x^2 - 16)

3. y = x^2 / (2x + 3)

4. y = x^2 / (x^2 + 4)

5. y = x^2 / (x^2 - 9)

6. y = 1 + x / (x^2 - 1)

7. y = x / (x^2 + 1)

8. y = (1 +x^2) / (1 - x^2)

9. y = (1 + x^2) / (1 - x^2)

10. y = (2x +3) / (x^2 - 4)

Thanks again!

EXAMPLE:
1. y = 2 / (x^2 +3)
Let; u = 2, v= x^2+3
u'= 0, v'=2x
y'=2x(2)/x^2+3)^2 <-----Use quotient rule.
y'=4x/(x^2+3)^2
Let y'=0 <---- stationary points occur when the tangent function to a curve is at 0
Thus, x=0 and y= 2/3 (Sub x value into original equation)

Types of Stationary points:
- Rising point of Inflexion - Before the stationary point, the function is positive and after the stationary point the function is positive.
- Falling point of inflexion- same as above just negative.
- Local minimum- negative before stationary point, after is positive.
- Local maximum- positive before stationary point, negative after.

For concavity try the double derivative.

Behaviour of functions as x-> infinity and x-> negative infinity.
asymptotes are anywhere where an undefined functional output occurs ie. 1/0 (vertical asymptote)