Question

HELP :( LINEARIZATION: Please use h in place of \Delta x and dx
I. If y= f(x)= 8 x^2+ 31 x + 12 find

a. dy =16xh+31
b. \Delta y = 8*(x+h)^2+31*(x+h)-8x^2-31x
II. Evaluate each of these quantities if x=10 and h=0.04

a. dy= 37.4
b. \Delta y = 7.6528

Answer 1

why are my answers wrong? :(

Answer 2

for part a) you should have dy =16xh+31h

Answer 3

why have 31 h?

Answer 4

if you start with
y= x
d (y = x) is
dy = dx
or using h instead of dx
dy = h

for your problem
\[ d (y = 8 x^2+ 31 x + 12 )\\
dy = 8 dx^2 + 31 dx \\
dy = 16x dx +31 dx \\
dy = 16x \ h + 31 \ h
\]

Answer 5

i thought we leave the constant alone?

Answer 6

oh because it was an x because finding the derivative?

Answer 7

compared with
\[ d (y = 31 x ) \\
dy = 31\ dx \]

Answer 8

yes it ='s to 1

Answer 9

\[ \frac{d}{dx} (y = 31 x) \\
\frac{d}{dx}y = 31 \frac{d}{dx}x\\
\frac{dy}{dx}= 31\frac{dx}{dx}\\
\frac{dy}{dx}= 31\cdot 1=31 \]

Answer 10

right.

Answer 11

but i was confused why i couldn't leave 31 alone

Answer 12

and why i had to add h in the end..

Answer 13

maybe a better way to think of it is

\[ \Delta y = \frac{dy}{dx} \Delta x \]
where dy/dx is the derivative of your function
\[ \frac{dy}{dx}= 16x + 31\]
and
\[ \Delta y = \frac{dy}{dx} \Delta x =(16x + 31)\Delta x= (16x + 31)h \]

Answer 14

oh okay, got cha.

Answer 15

what if it had like an x and a y inside

Answer 16

could i still apply this?

Answer 17

it would be the same idea, but you use "partial derivatives"

Answer 18

ex: dy/dx

Answer 19

so (d/dx+dy/dx)h ?

Answer 20

do you have a specific question?

Answer 21

it is a new topic to talk about functions of two variables f(x,y) = x^2 + y^2 for example
you won't see those until calculus II

Answer 22

well i know that we have to do this: 2x*y(dy/dx) +2y*(d/dx).. right?

Answer 23

are you asking about
f(x,y) = x^2 + y^2
or
z = x^2 + y^2
?
we find the partial derivative with respect to x (that means treat y as a constant)
fx (means partial derivative with respect to x) = 2x
fy = 2y
we then say
\[ \Delta z = 2x \Delta x + 2y \Delta y \]

Answer 24

But you won't see those type of problems.

Answer 25

oh okay, thank you so much! :)