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Math Help - Tangent lines and derivatives

  1. #1
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    Tangent lines and derivatives

    The Question asks:

    "Which of the following is an equation of the line tangent to the graph of
    f(x)= x^4 + 2x^2 at the point where f'(x)=1? "

    I found the derivative/slope to be:
    f'(x)= 4x^3 + 4x
    And since f'(x) = 1, I set it equal to one.

    1=4x^3 + 4x

    Even if I was able to factor that out (which I can't figure out how) I don't know how to find the equation of the tangent line without a point to work from.
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  2. #2
    Member Traveller's Avatar
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    Suppose you have factored it and x comes out to be some x'. Then what you want is a line that passes through (x', f(x')) with slope 1. Clearly, this line is unique. Consider the usual form of the equation of a line: y = mx + c. Here m is the slope and c is the intercept of the line with the y axis. You already have m=1, x=x', y=f(x'). So you can solve for c to get the equation.
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  3. #3
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    Quote Originally Posted by Traveller View Post
    Suppose you have factored it and x comes out to be some x'. Then what you want is a line that passes through (x', f(x')) with slope 1. Clearly, this line is unique. Consider the usual form of the equation of a line: y = mx + c. Here m is the slope and c is the intercept of the line with the y axis. You already have m=1, x=x', y=f(x'). So you can solve for c to get the equation.
    m=1
    y-f(x')= 1(x - x')
    y = x- x' + f(x')
    So c= x' + f(x') ?

    Our answer choice are:

    a. y = 8x - 5
    b. y= x + 7
    c. y= x + .763
    d. y= x - .122
    e. y= x - 2.146

    It can't be a. because m=1.
    But using the notation you gave me, all I get is a bunch of variables for c.
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  4. #4
    Member Traveller's Avatar
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    Your last step was wrong; c = f(x') - x'.

    See if you can solve for x from the derivative by factorizing. If you can't do that, you can always use Cartan's formula. x will come out to be something around 0.23673290386456308 ( I got it from here : Cubic equation Calculator - Calculation ). Find y and see which answer holds.
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