How do you find the tangent line to the curve #y=x^3-9x# at the point where #x=1#?
Given:
Let
So, the point we are targeting is
To find the slope of the tangent line there, we must differentiate
At So, the slope of the tangent line is Now, we use the point-slope formula to compute the equation, that is, Therefore, we get, A graph shows it:
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To find the tangent line to the curve y=x^3-9x at the point where x=1, we need to find the slope of the curve at that point.
First, we find the derivative of the function y=x^3-9x with respect to x. The derivative of x^3 is 3x^2, and the derivative of -9x is -9. Therefore, the derivative of the function is dy/dx = 3x^2 - 9.
Next, we substitute x=1 into the derivative to find the slope at that point. dy/dx = 3(1)^2 - 9 = -6.
The slope of the tangent line is -6.
To find the equation of the tangent line, we use the point-slope form of a line. The equation is y - y1 = m(x - x1), where (x1, y1) is the point on the curve where x=1, and m is the slope.
Substituting x=1 and y=1^3-9(1) into the equation, we have y - (-8) = -6(x - 1).
Simplifying, we get y + 8 = -6x + 6.
Rearranging the equation, we have y = -6x - 2.
Therefore, the equation of the tangent line to the curve y=x^3-9x at the point where x=1 is y = -6x - 2.
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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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