How do you find the points that the line tangent to #f(x)=x/(x+2)# has a slope 1/2?

Answer 1

The two points are (0,0) and (-4,2).

Known fact- f'(x)/gradient = 1/2.

Thought process- what are some possible values of (x,y) when f'(x) = 1/2?

f(x) = 1 - 2/(x+2) (By long division)

f'(x) = 0 - (-1)#(x+2)^-2#(2) = 2/#(x+2)^-2#
f'(x) = 1/2 1/2 = 2/#(x+2)^-2# #(x+2)^2# = 4 #(x+2)# = 2 or -2 #x = 0 or -4 #

When x = 0 f(0) = 0

when x = -4 f(-4) = 2

The two points are (0,0) and (-4,2).

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Answer 2

To find the points where the line tangent to the function f(x) = x/(x+2) has a slope of 1/2, we need to find the derivative of the function and set it equal to 1/2.

First, we find the derivative of f(x) using the quotient rule:

f'(x) = [(x+2)(1) - x(1)] / (x+2)^2

Next, we set the derivative equal to 1/2 and solve for x:

[(x+2)(1) - x(1)] / (x+2)^2 = 1/2

Simplifying the equation, we get:

2(x+2) - 2x = (x+2)^2

Expanding and simplifying further:

2x + 4 - 2x = x^2 + 4x + 4

Rearranging the terms:

x^2 + 4x + 4 - 2x - 2x - 4 = 0

Simplifying:

x^2 - 2x = 0

Factoring out x:

x(x - 2) = 0

Setting each factor equal to zero:

x = 0 or x - 2 = 0

Solving for x, we get:

x = 0 or x = 2

Therefore, the points where the line tangent to f(x) = x/(x+2) has a slope of 1/2 are (0, 0) and (2, 2/4).

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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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