What is the equation of the line with slope #3# which is tangent to the curve #f(x)=7xx^2#?
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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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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:
 Find the derivative of the function ( f(x) ) to determine the slope of the tangent line at any point on the curve.
 Set the derivative equal to the given slope, which is 3, to find the xcoordinate of the point of tangency.
 Substitute the xcoordinate of the point of tangency into the original function to find the corresponding ycoordinate.
 Use the pointslope form of the equation of a line to write the equation of the tangent line.
Let's proceed with these steps:

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

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

Substitute ( x = 2 ) into the original function to find the corresponding ycoordinate: [ f(2) = 7(2)  (2)^2 = 14  4 = 10 ]
So, the point of tangency is ( (2, 10) ).
 Use the pointslope 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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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