How to find (f) if f′(x)=16x^3+6x+7 and f(1)=−1 ?

3 Answers
Apr 4, 2018

#f(x) = 4x^4 + 3x^2 + 7x - 15#

Explanation:

We start by integrating both sides.

#f(x) = int 16x^3 + 6x + 7 dx#

#f(x) = 4x^4 + 3x^2 + 7x + C#

Now we solve for #C#.

#-1 = 4(1)^4 + 3(1)^2 + 7(1) + C#

#-1 = 4 + 3 + 7 + C#

#-1 - 14 = C#

#C = -15#

Hopefully this helps!

Apr 4, 2018

#f(x)=4x^4+3x^2+7x-15#

Explanation:

Integrate:
#16x^3# becomes #4x^4#
#6x# becomes #3x^2#
#7# becomes #7x#

So #f(x)# is #4x^4+3x^2+7x+C#.
Plug in #x=1#:

#4(1^4)+3(1^2)+7(1)+C#
#=4+3+7+C#
#=14+C#

Set #14+C# equal to #-1#:
#-1=14+C#

Solve for C:
#-15=C#

So your #f(x) = 4x^4 + 3x^2+7x-15#

Apr 4, 2018

#f(x)=4x^4+3x^2+7x-15#

Explanation:

We got:

#f'(x)=16x^3+6x+7#

#f(1)=-1#

And so,

#f(x)=intf'(x) \ dx#

#=int16x^3+6x+7 \ dx#

#=4x^4+3x^2+7x+C#

Therefore,

#4(1)^4+3(1)^2+7*1+C=-1#

#4*1+3*1+7+C=-1#

#4+3+7+C=-1#

#14+C=-1#

#C=-15#

So, the original function #f(x)# is:

#color(blue)(f(x)=barul|4x^4+3x^2+7x-15|)#