How do you find the equation of the line tangent to the graph of #y = x^2 - 3# at the point P(2,1)?

Answer 1

#4x-y=7#

The slope of a tangent to #y=x^2-3# is given by its derivative: #color(white)("XXXX")m=(dy)/(dx) = 2x#
At #(x,y) = (2,1)# the slope becomes (substituting #2# for #x# #color(white)("XXXX")m = 4#
The general slope point form for a line with slope #m# through a point #(hatx,haty)# is #color(white)("XXXX")y-haty = m(x-hatx)#
Substituting #m=4#, #hatx=2#, and #haty=1# #color(white)("XXXX")y-1 = 4(x-2)#
This could be re-written in standard form as #color(white)("XXXX")4x-y = 7#
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Answer 2

To find the equation of the line tangent to the graph of y = x^2 - 3 at the point P(2,1), we can use the concept of differentiation.

First, we differentiate the given function y = x^2 - 3 with respect to x to find its derivative.

The derivative of y = x^2 - 3 is dy/dx = 2x.

Next, we substitute the x-coordinate of the point P(2,1) into the derivative to find the slope of the tangent line at that point.

Substituting x = 2 into dy/dx = 2x, we get the slope m = 2(2) = 4.

Now, we have the slope of the tangent line, and we also have a point on the line, P(2,1). We can use the point-slope form of a linear equation to find the equation of the tangent line.

Using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the point (2,1) and m is the slope 4, we substitute the values into the equation.

Therefore, the equation of the line tangent to the graph of y = x^2 - 3 at the point P(2,1) is y - 1 = 4(x - 2).

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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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