How do you find the equation of the line tangent to the graph of #y=x^2# at x=2?

Answer 1

I found: #y=4x-4#

First you need the slope #m# of the tangent. You find it evaluating the derivative of your function in #x=2#:
#y'=2x#
so #m=y'(2)=2*2=4# this is the slope of the tangent at the point of coordinates:
#x_0=2# and #y_0=x^2=2^2=4#;
now you can use the relationship:
#y-y_0=m(x-x_0)#
#y-4=4(x-2)# rearranging:
#y=4x-8+4#
#y=4x-4#

Graphically:

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Answer 2

To find the equation of the line tangent to the graph of y=x^2 at x=2, we can use the concept of differentiation.

First, we need to find the derivative of the function y=x^2 with respect to x. The derivative of x^2 is 2x.

Next, substitute x=2 into the derivative to find the slope of the tangent line at x=2. When x=2, the slope is 2(2) = 4.

Now, we have the slope of the tangent line, which is 4. To find the equation of the line, we also need a point on the line. Since the line is tangent to the graph of y=x^2 at x=2, we can use this point as our reference.

At x=2, the corresponding y-value on the graph is y=2^2 = 4. So, the point (2, 4) lies on the tangent line.

Using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept, we can substitute the values we have:

4 = 4(2) + b

Simplifying, we get:

4 = 8 + b

Subtracting 8 from both sides, we find:

b = -4

Therefore, the equation of the line tangent to the graph of y=x^2 at x=2 is y = 4x - 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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