# What is the equation of the line tangent to # f(x)=(e^x-x)^2 # at # x=3#?

We must first distinguish the function,

at the third point

Consequently, our tangent line equals

We now need to use the point where x=3 to solve for c.

thus, this is our tangent line:

which, given that e is irrational, can be roughly equal to

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To find the equation of the line tangent to the function f(x) = (e^x - x)^2 at x = 3, we need to find the derivative of the function and evaluate it at x = 3.

First, let's find the derivative of f(x) using the chain rule:

f'(x) = 2(e^x - x)(e^x - 1)

Next, we substitute x = 3 into the derivative to find the slope of the tangent line at x = 3:

f'(3) = 2(e^3 - 3)(e^3 - 1)

Now, we have the slope of the tangent line. To find the equation of the line, we need a point on the line. Since the line is tangent to the function at x = 3, we can use the point (3, f(3)).

To find f(3), substitute x = 3 into the original function:

f(3) = (e^3 - 3)^2

Now we have the slope and a point on the line. We can use the point-slope form of a linear equation to find the equation of the tangent line:

y - f(3) = f'(3)(x - 3)

Simplifying this equation will give us the final equation of the line tangent to f(x) at x = 3.

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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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