How do you find the equation of the line tangent to the graph of #f(x) = 6 - x^2# at x = 7?

Answer 1

#y = -14x+55#

Given -
#y = 6-x^2#
It is a quadratic function.
Since co-efficient of #x^2# is negative, it is an downward facing 'U'shaped curve.

The slope of the curve at any given point is its 1st erivative.

#dy/dx=-2x#

At# x = 7 #; the slope of the curve is #m = -2(7) = -14#

At #x=7#; the y-co-ordinate of the curve is -

#y=6-(7^2)=6-49= -43#

#(7, -43) # is a point on the curve and on the tangent.
The slope of the tangent is -14

Equation of the tangent is
# mx +c=y#
#(-14).7+c=-43#
#-98+c= -43#
#c = -43 +98 = 55#

#y = -14x+55#

Refer the graph

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Answer 2

To find the equation of the line tangent to the graph of f(x) = 6 - x^2 at x = 7, we need to find the derivative of the function f(x) and evaluate it at x = 7. The derivative of f(x) is f'(x) = -2x. Evaluating f'(x) at x = 7, we get f'(7) = -2(7) = -14.

The slope of the tangent line is equal to the derivative of the function at the given point, so the slope of the tangent line is -14.

To find the equation of the tangent line, we use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in the values x1 = 7, y1 = f(7) = 6 - (7^2) = -43, and m = -14, we have y - (-43) = -14(x - 7). Simplifying this equation gives the equation of the tangent line as y = -14x + 85.

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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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