How do you find the equation of the line tangent to #f(x) = x^2 + 2x +1#, with the point (3, 4)?
y + 4x + 8 = 0
The equation of the tangent is : y  b = m(x  a ) where m
represents the gradient and (a , b ) a point on the line.
Require to find m , (a , b ) is given (3 , 4 ) Now m = f'(x).
f'(x) = 2x + 2 and f'(3) = 2(3) + 2 =  6 + 2 =  4 = m
Equation is : y  4 = 4 ( x + 3 )
ie y  4 =  4x  12
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To find the equation of the line tangent to the function f(x) = x^2 + 2x + 1 at the point (3, 4), we need to find the derivative of the function and evaluate it at x = 3.
The derivative of f(x) = x^2 + 2x + 1 is f'(x) = 2x + 2.
To find the slope of the tangent line, we substitute x = 3 into the derivative: f'(3) = 2(3) + 2 = 4.
The slope of the tangent line is 4.
Using the pointslope form of a linear equation, y  y1 = m(x  x1), where (x1, y1) is the given point and m is the slope, we substitute (3, 4) and 4 into the equation:
y  4 = 4(x  (3)).
Simplifying, we get the equation of the tangent line: y  4 = 4(x + 3).
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To find the equation of the tangent line to the function ( f(x) = x^2 + 2x + 1 ) at the point ((3, 4)), follow these steps:

Find the derivative of the function ( f(x) ) using the power rule. The derivative of ( f(x) ) is ( f'(x) = 2x + 2 ).

Evaluate the derivative at the xcoordinate of the given point (3) to find the slope of the tangent line. ( f'(3) = 2(3) + 2 = 4 ).

Use the pointslope form of the equation of a line to write the equation of the tangent line. The pointslope form is ( y  y_1 = m(x  x_1) ), where ( (x_1, y_1) ) is the given point and ( m ) is the slope.

Substitute the values of the given point and slope into the pointslope form. ( y  4 = 4(x + 3) ).

Simplify the equation to obtain the equation of the tangent line. ( y  4 = 4x  12 ). This simplifies to ( y = 4x  8 ).
Thus, the equation of the tangent line to ( f(x) = x^2 + 2x + 1 ) at the point ((3, 4)) is ( y = 4x  8 ).
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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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