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.
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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