# If the line 4x-9y=0 is tangent in the first quadrant to the graph of y =1/3x^3+c what is c?

We'll have to do some thinking in order to solve this problem.

Note that we take the positive square root because the point in question is in the first quadrant, as specified by the question.

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To find the value of c, we need to determine the point of tangency between the line 4x-9y=0 and the graph of y = 1/3x^3+c.

First, we need to find the derivative of the equation y = 1/3x^3+c, which is y' = x^2.

Next, we substitute the equation of the line into the derivative to find the x-coordinate of the point of tangency: 4x - 9y = 0 → 4x - 9(1/3x^3+c) = 0 → 4x - 3x^3 - 9c = 0.

Now, we solve this equation for x.

By factoring, we get: x(4 - 3x^2) = 9c.

Since the line is tangent in the first quadrant, the x-coordinate of the point of tangency must be positive.

Setting x = 0, we find that x = 0 is not a solution.

Setting 4 - 3x^2 = 0, we find that x = √(4/3) is a solution.

Therefore, the x-coordinate of the point of tangency is x = √(4/3).

To find the corresponding y-coordinate, we substitute this value of x into the equation y = 1/3x^3+c:

y = 1/3(√(4/3))^3 + c = 2√3/9 + c.

Since the line is tangent to the graph, the y-coordinate of the point of tangency must be equal to 0.

Setting 2√3/9 + c = 0, we find that c = -2√3/9.

Therefore, c = -2√3/9.

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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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