# For what values of a and b will the parabola y=x^2 + ax + b be tangent to the curve y=x^3 at point (1,1)?

The values are

In order for two curves to be tangent to each other at a given point, they must have the same tangent line at that point. (If their tangent lines at the given point were different, the two curves would "pass through" each other rather than just "graze" each other.)

Thus, the slope-point form of the tangent line required is:

These will give us our two equations in two unknowns. The first one gives us

and the second one gives us

Here is a graph of the two curves: graph{(y-x^3)(y-x^2-x+1)=0 [-4.937, 4.93, -1.716, 3.22]}

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The parabola y=x^2 + ax + b will be tangent to the curve y=x^3 at point (1,1) when the values of a and b satisfy the following conditions: a = 3 and b = -1.

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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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