Results 1 to 7 of 7

Math Help - [SOLVED] Inverse functions

  1. #1
    Ife
    Ife is offline
    Member Ife's Avatar
    Joined
    May 2009
    From
    Trinidad & Tobago
    Posts
    119

    [SOLVED] Derivatives of functions

    Suppose that g(0) = -7 and that g'(t)=5 for all of t. Must g (t) = 5t-7 for all of t??
    Last edited by Ife; December 13th 2009 at 12:45 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Quote Originally Posted by Ife View Post
    Suppose that g(0) = -7 and that g'(t)=5 for all of t. Must g (t) = 5t-7 for all of t??
    Absolutely not. There are an infinte number of functions that pass through the point (0,-7) having slope 5.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Ife
    Ife is offline
    Member Ife's Avatar
    Joined
    May 2009
    From
    Trinidad & Tobago
    Posts
    119
    Quote Originally Posted by VonNemo19 View Post
    Absolutely not. There are an infinte number of functions that pass through the point (0,-7) having slope 5.
    Great. I had the right answer, but i am not sure that i had the right thought process... i had said something along the lines of: If y= 2x-7, then y^{-1} = \frac {x+7}{2}. when x=0, y=-7, but when y^{-1} = 5, x = -2 so that negates the question. Is my thought process skewed? Or is that a valid approach?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Ife View Post
    Suppose that g(0) = -7 and that g'(t)=5 for all of t. Must g (t) = 5t-7 for all of t??
    Yes:

    \frac{dg}{dt} = 5 \Rightarrow g(t) = 5t + C. g(0) = 7 \Rightarrow 7 = C. Therefore g(t) = 5t + 7.

    Quote Originally Posted by VonNemo19 View Post
    Absolutely not. There are an infinte number of functions that pass through the point (0,-7) having slope 5.
    Correct. But not having a slope 5 for all values of t ...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Ife
    Ife is offline
    Member Ife's Avatar
    Joined
    May 2009
    From
    Trinidad & Tobago
    Posts
    119
    Quote Originally Posted by mr fantastic View Post
    Yes:

    \frac{dg}{dt} = 5 \Rightarrow g(t) = 5t + C. g(0) = 7 \Rightarrow 7 = C. Therefore g(t) = 5t + 7.


    Correct. But not having a slope 5 for all values of t ...
    Thanks, I was reviewing the question yesterday and realised that entire thing before was Wrong! Because we are not looking at the inverse, but rather, the derivative! I was about to make a note of that here. Didn't see you had responded.

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Ife
    Ife is offline
    Member Ife's Avatar
    Joined
    May 2009
    From
    Trinidad & Tobago
    Posts
    119
    Quote Originally Posted by VonNemo19 View Post
    Absolutely not. There are an infinte number of functions that pass through the point (0,-7) having slope 5.
    This is if we are looking at the inverse function but we are not. The question asked about the derivative! We had the totally wrong idea here... I hope this post doesn't mislead anyone...
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,240
    Thanks
    1795
    Quote Originally Posted by Ife View Post
    Suppose that g(0) = -7 and that g'(t)=5 for all of t. Must g (t) = 5t-7 for all of t??
    Yes. By the mean value theorem, two functions that have the same derivative must differ by a constant:
    Let h(x)= f(x)- g(x). f'(t)= g'(t), then h'(x)= 0 for all t. By the mean value theorem, h(b)- h(a)/(b- a)= 0 so h(b)- h(a)= 0 and h(b)- h(a)= 0. That is, h is a constant and so f and g differ by a constant.

    Since f(t)= 5t has f'(x)= 5 for all t,any function with derivative always equal to 5 must be of the form g(x)= 5x+ C. In order that g(0)= -7, C must be -7.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] inverse functions
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: September 28th 2009, 06:59 AM
  2. [SOLVED] help with inverse trig functions
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: April 6th 2009, 03:08 PM
  3. [SOLVED] inverse functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 25th 2009, 04:59 PM
  4. [SOLVED] Inverse Trigonometric Functions
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: March 17th 2009, 01:32 PM
  5. [SOLVED] Inverse Functions
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 19th 2008, 02:35 PM

Search Tags


/mathhelpforum @mathhelpforum