# How do you find the equation of the line tangent to the graph of #f(x)= x(sqrt(x)-1)# at the value x=4?

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To find the equation of the line tangent to the graph of f(x) = x(sqrt(x)-1) at x = 4, we need to find the slope of the tangent line and the point of tangency.

First, we find the derivative of f(x) using the product rule and simplify it: f'(x) = (sqrt(x) - 1) + x(1/2)(1/sqrt(x)) f'(x) = sqrt(x) - 1 + (x/2sqrt(x)) f'(x) = sqrt(x) + (x/2sqrt(x)) - 1

Next, we substitute x = 4 into f'(x) to find the slope of the tangent line at x = 4: f'(4) = sqrt(4) + (4/2sqrt(4)) - 1 f'(4) = 2 + (4/4) - 1 f'(4) = 2 + 1 - 1 f'(4) = 2

So, the slope of the tangent line at x = 4 is 2.

To find the point of tangency, we substitute x = 4 into f(x): f(4) = 4(sqrt(4) - 1) f(4) = 4(2 - 1) f(4) = 4(1) f(4) = 4

Therefore, the point of tangency is (4, 4).

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 found: y = 2x + b

To find b, we substitute the coordinates of the point of tangency (4, 4): 4 = 2(4) + b 4 = 8 + b b = -4

Thus, the equation of the line tangent to the graph of f(x) = x(sqrt(x)-1) at x = 4 is y = 2x - 4.

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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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