How do you find the slope and the equation of the tangent line to the graph of the function at the given value of x for #f(x)=x^4-13x^2+36# at x=2?

Answer 1

The slope is #-20#, the equation is #y=-20x+40#.

First of all, the slope is given by the derivative, so you have that

#f'(x) = 4x^3 - 26 x#, and thus #f'(2)= 32-52=-20#
Also, we know that the tangent line must pass across the point #2,f(2)#, namely #(2, 0)#
If you know that a line has slope #m#, and crosses the point #(x_0,y_0)#, then you obtain its equation with the formula
#y-y_0 = m(x-x_0)#. Plugging the values, we have
#y=-20(x-2)#, which we can rearrange into #y=-20x+40#

As you can see in this link , the line is actually tangent in the requested point.

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Answer 2

To find the slope of the tangent line to the graph of the function at x=2, we need to find the derivative of the function and evaluate it at x=2. The derivative of f(x)=x^4-13x^2+36 is f'(x)=4x^3-26x. Evaluating f'(x) at x=2, we get f'(2)=4(2)^3-26(2) = 32-52 = -20. Therefore, the slope of the tangent line at x=2 is -20.

To find the equation of the tangent line, we use the point-slope form of a line, y-y1=m(x-x1), where m is the slope and (x1, y1) is a point on the line. We already have the slope, -20, and the point (2, f(2)). Plugging these values into the equation, we get y-f(2)=-20(x-2). Simplifying, we have y-f(2)=-20x+40. Rearranging the equation, we get y=-20x+f(2)+40.

Therefore, the equation of the tangent line to the graph of the function f(x)=x^4-13x^2+36 at x=2 is y=-20x+f(2)+40.

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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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