# Assignment one Jase Smith

## Probelm one

Provide a clear description of the method of u-substitution for definite integrals (chapter 5.6, called “change of variables formula”) and demonstrate its use by computing the integral:
$$\int^2_0 \frac x{(1-x^2)^3}$$
first step:
let u= $1-x^2$
$\frac {du} {dx} = 2x$
$\frac {1} {2x}du= dx$

second step:
$\frac {x} {u^3} \times \frac {1} {2x}du$
cross out the two X's in the problem, that gives us
$\frac {1} {2U^3} du$
Then that simplifies to
$\frac{1}{2} u^{-3} du$
Then you take the antideravtive of $u^{-3}$ that gives you
$\frac {1}{2}(-\frac{1}{2})(u^{-2})$
from this put it all back together and you have
$$\int^2_0 -\frac{1}{4} u^{-2}$$
Step Three:
$$\int^2_0 -\frac{1}{4} (1-x^2)^{-2}$$
Then
$$\int^2_0 -\frac{1}{4 (1-x^2)^{-2}}$$
now find U
$u=1-x^2$$
$u=1-0^2$, $u=1$
$u=1-2^2$, $u=3$

And plug the U back into the equation

$$\frac{1}{4 (1-12^2)^{-2}} -\frac {1}{4(1-0^2)^2}$$
which ends up as
$$\frac {1}{36} - \frac{1}{4} = -\frac{2}{9}$$


## Problem Two

### Form 1

State the two forms of the fundamental theorem of
calculus and discuss the kinds of problems that
form helps us solve. Include at least one example for
each form.

- if G is an antidervative of f(G'(x)=f(x)) then $\int^a_b f(x)dx=G(b)-G(a)=G(x)|^b_a$

it would be approprate to use when finding the area under the curve of a function when graphed.

example:
$$\int^4_1 x^{1/2} dx = \frac{2}{3} x^{3/2}$$

$$\frac{2}{3}(4)^{3/2}-\frac{2}{3}(1)^{3/2}$$

$$ \frac {14}{3}$$

### Form 2
- if f(x) cont. on [a,b] define: $A(x) = \int^x_af(t)dt$. Then $A'(x) = f(x)$

this is used for when you arent sure what X is yet

Exapmle:
$$\frac {d}{dx}[\int^x_0 \sqrt{t^2+4} dt] $$

$$ f(t)= \sqrt{t^2+4}$$

$$\frac {d}{dx} [\int^x_0 f(t)dt]= \frac {d}{dx} [f(t)|^x_0]$$

$$\frac {d}{dx} [f(x)-f(0)]= f(x)-0 = f(x)$$

$$f(x)= \sqrt{x^2+4}$$

## Problem Three

please Provide the two forms of the limit definition of the derivative (both the x->a and the h->0 forms) and use those forms to directly compute the derivative of $f(x)=x^2+1$ at x=a (So you will be providing two ways of computing that derivative, one for each form of the limit).

### First form:

$$ lim_h->0 frac{f(a+h)-f(a)}{h}$$

with all x's filled in with (x+h) it turns to

$$\frac{(x+h)^2+1+(x^2+1)}{h}$$

Then through foiling and simplifying it goes to

$$\frac{2xh+h^2}{h}$$

Next through factoring out it turns to

$$lim_h->0(2x+h)$$

so as h->0 the derivative of $x^2+1$ is 2x

### Second form:
$$ f'(a)= lim_x->a \frac {f(x)-f(a)}{x-a}$$

i ran out of time typing this i am very sorry and should have started sooner i underestimated how long typing and coding would take but i can send you the written form if you'd like, and please excuse any spelling errors.