What is the equation of the line with slope #3# which is tangent to the curve #f(x)=7x-x^2#?

Answer 1

#y=3x+4#

If #f(x)=7x-x^2# then the slope (for any general #x# value) is #f'(x)=7-2x#
when the slope is #m=f'(x)=3# then #7-2x=3# #color(white)("XX")-2x=-4# #color(white)("XX")x=2#
If #x=2# then #color(white)("XXX")f(color(red)(2))=7*(color(red)(2))-color(red)(2)^2=14-4=color(blue)(10)#
and the point on the curve were #f'(x)=3# occurs at #(color(red)(2),color(blue)(10))#
Therefore, for the tangent, we have a slope of #color(green)m=3# and a point #(color(red)(2),color(blue)(10))#
Using the slope-point form of the equation: #color(white)("XXX")y-color(blue)(10)=color(green)(3)(x-color(red)(2))#
or #color(white)("XXX")y=3x+4#
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Answer 2

The equation of the line with slope 3 which is tangent to the curve (f(x) = 7x - x^2) is (y = 3x + 7).

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Answer 3

To find the equation of the line that is tangent to the curve ( f(x) = 7x - x^2 ) and has a slope of 3, you need to follow these steps:

  1. Find the derivative of the function ( f(x) ) to determine the slope of the tangent line at any point on the curve.
  2. Set the derivative equal to the given slope, which is 3, to find the x-coordinate of the point of tangency.
  3. Substitute the x-coordinate of the point of tangency into the original function to find the corresponding y-coordinate.
  4. Use the point-slope form of the equation of a line to write the equation of the tangent line.

Let's proceed with these steps:

  1. Find the derivative of the function ( f(x) = 7x - x^2 ): [ f'(x) = \frac{d}{dx} (7x - x^2) = 7 - 2x ]

  2. Set the derivative equal to the given slope: [ 7 - 2x = 3 ] [ -2x = 3 - 7 ] [ -2x = -4 ] [ x = 2 ]

  3. Substitute ( x = 2 ) into the original function to find the corresponding y-coordinate: [ f(2) = 7(2) - (2)^2 = 14 - 4 = 10 ]

So, the point of tangency is ( (2, 10) ).

  1. Use the point-slope form of the equation of a line to write the equation of the tangent line: [ y - y_1 = m(x - x_1) ] [ y - 10 = 3(x - 2) ] [ y - 10 = 3x - 6 ] [ y = 3x + 4 ]

Therefore, the equation of the line with slope 3 that is tangent to the curve ( f(x) = 7x - x^2 ) is ( y = 3x + 4 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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