How do you find the equation of a line tangent to the function #y=3x-4sqrtx# at x=4?

Answer 1

#y=2x-4#

Find the point first where the tangent line will intersect on the curve:

#y(4)=3(4)-4sqrt4=4#
Implying the point #(4,4)#.
To find the slope of the tangent line, take the derivative of the function first, using the power rule: #d/dxx^n=nx^(n-1)#
#y=3x^1-4x^(1/2)#
#dy/dx=3(1)x^0-4(1/2)x^(-1/2)#
#dy/dx=3-2/sqrtx#
So, the slope at #x=4# is:
#dy/dx|_(x=4)=3-2/sqrt4=2#
Using the point #(4,4)# and slope #2# to write the tangent line:
#y-y_0=m(x-x_0)#
#y-4=2(x-4)#
#y=2x-4#

graph{(y-3x+4sqrtx)(y-2x+4)=0 [-1, 10, -5, 16]}

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Answer 2

To find the equation of a line tangent to the function y=3x-4sqrt(x) at x=4, we need to find the slope of the tangent line at that point and then use the point-slope form of a linear equation.

To find the slope of the tangent line, we can take the derivative of the function y=3x-4sqrt(x) with respect to x.

The derivative of y=3x-4sqrt(x) is dy/dx = 3 - 2/sqrt(x).

Substituting x=4 into the derivative, we get dy/dx = 3 - 2/sqrt(4) = 3 - 2/2 = 3 - 1 = 2.

So, the slope of the tangent line at x=4 is 2.

Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute x1=4, y1=3(4)-4sqrt(4) = 12-8 = 4 into the equation.

Therefore, the equation of the line tangent to the function y=3x-4sqrt(x) at x=4 is y - 4 = 2(x - 4).

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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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