Problem involving differentials

temporaryinsanit

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Please Help With This Question:

:?:
Let f be a function with f(1) = 4 such that for all points (x,y) on the graph of f the slope is given by 3x^2 + 1 / 2y

A) Find the slope of the graph of f at the point where x = 1

B) Write an equation for the line tangent to the graph of f at x = 1 and use it to approximate f (1.2)

C) Find f(x) by solving the separable differential equation dy/dx = 3x^2 / 2y with the initial condition f(1) = 4

Thank You
 
part C:

dydx=3x22y\displaystyle \frac{dy}{dx}=\frac{3x^{2}}{2y}

Separate variables:

2ydy=3x2dx\displaystyle 2\int y dy=3\int x^{2}dx

y2=x3+C\displaystyle y^{2}=x^{3}+C

Now, use the initial condition of find C.
 
\(\displaystyle Here \ is \ the \ graph \ of \ f(x), \ f(1)=4 \ of \ your \ first \ question. \ I \ graphed \ it \ numerically \ on \ Maple \ 8 \\)

as it is impossible to obtain a solution to this differential equation in closed form.2nd plot\displaystyle as \ it \ is \ impossible \ to \ obtain \ a \ solution \ to \ this \ differential \ equation \ in \ closed \ form. 2nd \ plot

[attachment=1]def.jpg[/attachment

The slope at (1,4) is about m = 3.125, hence y =˙ 3.125x+.875 (red line), and f(1.2) = 4.751.\displaystyle The \ slope \ at \ (1,4) \ is \ about \ m \ = \ 3.125, \ hence \ y \ \dot= \ 3.125x+.875 \ (red \ line), \ and \ f(1.2) \ = \ 4.751.

Also note, since f(x) is increasing, then f(x) must be less than zero, which it is  see 1st plot.\displaystyle Also \ note, \ since \ f(x) \ is \ increasing, \ then \ f'(x) \ must \ be \ less \ than \ zero, \ which \ it \ is \ - \ see \ 1st \ plot.
 

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