Explore BrainMass

Explore BrainMass

    Equation of a straight line which is tangent to a curve

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1.
    See attachment for diagram.

    © BrainMass Inc. brainmass.com December 24, 2021, 4:45 pm ad1c9bdddf
    https://brainmass.com/math/derivatives/equation-straight-line-which-tangent-curve-5295

    Attachments

    SOLUTION This solution is FREE courtesy of BrainMass!

    For the curve f(x) = x - 1/3x^2 (one third x squared), find the equation of the straight line which is tangent to this curve at the point x = 1.

    Equation of the curve= f(x) = x - 1/3x^2
    y= x- 1/3 x ^2
    Finding derivative is easy
    Derivative of x ^n is n x ^ (n-1)
    Thus derivative of x ^2 is
    2 x ^(2-1)= 2 x ^ 1 that is 2x
    Derivative of x is
    x= x ^1
    therefore derivative of x is 1 * x ^ 1-1
    = 1 * X ^ 0
    =1 ( as X^0=1)
    Similarly derivative of 1/x^2 can be solved
    1/x^2 can be written as x ^ -2
    Derivative of x ^-2 is -2 * x ^ (-2-1)= -2 x ^ -3 = -2/ x ^3

    If the value is multiplied/ divided by a constant multiply/ divide the derivative by the constant.
    Thus derivative of x ^2 is 2x.
    Derivative of 3 x ^2 is 3 *( 2 x)= 6x

    Therefore derivative of the curve f(x)= y= x- 1/3 x ^2
    dy/dx = 1- 1/3 (-2/ x ^3)=1+ 2/3 (1/x^3)

    At x=1
    dy/dx= 1+ 2/3(1/1^3)=1+2/3= 5/3

    The y coordinate of the point on the curve where x=1 is
    y= x- 1/3 x ^2
    =1-1/3(1/1^2)=1-1/3=2/3 ( as you have rightly calculated)
    Thus the coordinates through which the straight line passes is (1, 2/3)
    Since the st line is tangent to this curve at x=1 the slope of the st line = slope of the curve we have calculated. = 5/3
    Equation of a st line is y= mx+c
    m= 5/3
    Substituting the values of m, y, x in the equation we can calculate the value of c
    y= mx+c
    2/3=5/3 (1)+ c ( since the st line passes through (1, 2/3)
    or 2/3=5/3 + c or c= 2/3-5/3=-3/3=-1
    Therefore the equation of the st line is
    y= 5/3 (x) - 1

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 4:45 pm ad1c9bdddf>
    https://brainmass.com/math/derivatives/equation-straight-line-which-tangent-curve-5295

    ADVERTISEMENT