Couple of simple (I hope) questions

[parabol]

Homework Statement

1st problem

I'm struggling to answer this question using my calculus workbook.

Find the point of intersection of the lines:

$$y_{1}=3x+2$$
$$3y_{2}=16-x$$

I can solve this question as a simultaneous equation, but, as it is in my calculus question paper believe it to be less simple. Is there some method of differentiating the equations in order to work out the intersect point?

2nd problem

If $$f(x)=7x^3 + 8x^2 - x + 11$$, evaluate;

a) $$\int\stackrel{+1}{-1}f(x)dx$$

b) $$\int\stackrel{+1}{-1}f ' (x)dx$$

c) $$\int\stackrel{+1}{-1}f '' (x)dx$$

I'm happy with the integration but just wanted to make sure that I am working the integration of first the function, then the higher order derivatives or should it be higher order integration.

Sorry if the problems seem simple but I'm struggling to understand my workbook.

 

[Elucidus]

Your first exercise is just a system of simultaneous equations as you suspected. Differentiation is not needed (and wouldn't help as two functions with equal derivative are not necessarily equal).

The second problem is simpler than it might appear. The problem deals with integrals of higher-order derivative. The Fundamental Theorem of Calculus gives you a handle on this one.

$$\text{If } F'(x) = f(x) \text{ then } \int{f(x)\;{dx}} = F(x) + C$$

--Elucidus

 

[parabol]

Thanks for the reply.

Happy with the first part.

But for the 2nd. Are you saying that I have carried it out incorrectly by findging the first and second derivative and then integrating them and working out thorugh the boundary values?

 

[Elucidus]

But for the 2nd. Are you saying that I have carried it out incorrectly by findging the first and second derivative and then integrating them and working out thorugh the boundary values?

I am saying that $\int_{-1}^1 f'(x)\;dx=f(1)-f(-1)$ since f(x) is an antiderivative of f'(x). Patern continues...

--Elucidus

 

[parabol]

Thanks for your help. I reckon I have it sorted now.

Cheers