Question

Find the equation of the line tangent to the graph of the given function at the point with the indicated x-coordinate.


f(x) = (x^0.5 + 2)(x^2 + x); x = 1

Answer 1

how much have u tried?
where are u stuck?

Answer 2

to get the equation of the line tangent
u need slope of tangent, any ideas how to find slope of tangent when the function is given ?

Answer 3

set the equation equal to 0?

Answer 4

umm... its actually , finding the first derivative of the function at given point (x=1)
so what u get when u derivate f(x) = (x^0.5 + 2)(x^2 + x) ?

Answer 5

7.5

Answer 6

what was that ?
are u aware what is differentiation ?

Answer 7

am i supposed to subsitute 1 for x?

Answer 8

no, have u been taught differentiation ?

Answer 9

very minimal

Answer 10

what about product rule in differentiation ?
because u have to use product rule here

Answer 11

yes i do
dy/dx = f'(x)g(x) + f(x) g'(x)

Answer 12

good :)
so now put f(x) = (x^0.5 + 2) and g(x)=(x^2 + x)
and find dy/dx

Answer 13

(2.5)(x^0.5+2) + (x^2+x)(3) ?

Answer 14

umm, nopes
when f(x) = (x^0.5 + 2) , what is f'(x) = ?

Answer 15

not correct?

Answer 16

.5x+2?

Answer 17

nopes, derivative of x^n = n x^{n-1}
and derivative of constant = 0
so u get f'(x) = 0.5 x^{-0.5} +0
got this ?

Answer 18

i see, so it would be 0.5x^(-0.5) x (x^0.5 + 2)

Answer 19

f'(x)g(x) + f(x) g'(x)
= (0.5 x^{-0.5} )((x^2 + x) + f(x)g'(x)
now find g'(x)

Answer 20

sorry but i dont know how to find g'x

Answer 21

you know that derivative of x^n = n x^{n-1}
so what will be derivative of x^2 ?

Answer 22

2

Answer 23

what u get when u put n=2 in n x^{n-1}

Answer 24

2x

Answer 25

now its correct, and whats the derivative of x ?

Answer 26

1

Answer 27

correct so g'(x) = 2x+1
and entire derivative will be
f'(x)g(x) + f(x) g'(x)
= (0.5 x^{-0.5} )((x^2 + x) + (x^0.5 + 2)(2x+1)
do u understand till here ?
now just put x=1 in this , what u get dy/dx as ?

Answer 28

so i plug in 1 for every x?

Answer 29

yes ! and u get slope of line as dy/dx

Answer 30

oh ok

Answer 31

so what u got the slope as ?

Answer 32

i got 11.8

Answer 33

(0.5 x^{-0.5} )((x^2 + x) + (x^0.5 + 2)(2x+1)
= (0.5 1^{-0.5} )((1^2 + 1) + (1^0.5 + 2)(2(1)+1)
= (0.5)(2) + (1+2)(2+1)
= 1 + 3*3 = 10
got this ?

Answer 34

oh i found my mistake, is that the final answer?

Answer 35

or what is the final equation?

Answer 36

that is the slope of line, m=10
you need ans equation, you know how to find the equation of line when slope and point are given ?

Answer 37

for that u need y when x=1
y= f(x) = (x^0.5 + 2)(x^2 + x)
put x=1 here, what u get y as ?

Answer 38

so u get y= (1+2)(1+1) = 6
so the point is (x1,y1)= (1,6) and with m=10
use the point slope form of equation of line:
y-y1 = m (x-x1)
and u get the equation of line.

Answer 39

y-6 = 10(x-1) =

y = 10x+5

Answer 40

i mean y=10x-4

Answer 41

that is correct , y= 10x-4 is your final answer