How do you find the slope and tangent line to the curve #y=6-x^2# at x=7?

Answer 1

Slope #m=-14#
equation: #y=-14x+55#

To find the slope you need to derive your function and evaluate the derivative at #x=7#:
deriving:
#y'=-2x#

at #x=7#
#y'(7)=-14#

To find the equation of the tangent line you need also the value of #y# at #x=7#; by substituting into your function:
#y(7)=6-49=-43#
So basically your tangent has solpe #m=-14# and passes through #x_0=7# and #y_0=-43#;
Now use the relationship:
#y-y_0=m(x-x_0)# to find the equation of your line:
#y+43=-14(x-7)#
#y=-14x+55#

Graphicall (the red line is the tangent):

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Answer 2

To find the slope of the curve at a specific point, we need to find the derivative of the function. The derivative of y=6-x^2 is dy/dx = -2x.

To find the slope at x=7, we substitute x=7 into the derivative: dy/dx = -2(7) = -14.

Therefore, the slope of the curve at x=7 is -14.

To find the equation of the tangent line at x=7, we use the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the point on the curve.

Substituting x=7, y=6-7^2=-43 into the equation, we have: y - (-43) = -14(x - 7).

Simplifying, we get the equation of the tangent line: y = -14x + 105.

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Answer from HIX Tutor

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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