How do you find the equation of the tangent line to the graph of f(x) at given point if #f(x)= sqrt(x+1))# at (0,1)?

Answer 1

#y=1/2x +1#

#f(x) = sqrt(x+1) = (x+1)^(1/2)#
We are to find the tangent to #f(x)# at #(0,1)#
First lets confirm that #(0,1)# is a point on #f(x)#
#f(0) = (0+1)^(1/2) -> f(0) = 1#
Hence, #(0,1)# is a point on #f(x)#
Apply power and chain rules to #f(x)#
#f'(x) = 1/2(x+1)^(-1/2) * d/dx (x+1)#
# =1/(2sqrt(x+1)) * 1#
Now, the slope of #f(x)# at #x=0# is #f'(0)#
#f'(0) = 1/(2sqrt1) = 1/2#
The tangent to #f(x)# at #x=0# will have the form #y=mx+c# and from the above we know that #m=1/2#
Thus, #y =1/2x +c# at #(0,1)#
#:. c=1#
Hence, our tangent is : #y=1/2x +1# as we can see from the graphs below.

graph{(y-sqrt(x+1))(y-1/2x-1)=0 [-1.84, 2.487, -0.099, 2.064]}

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Answer 2

To find the equation of the tangent line to the graph of f(x) at the given point (0,1), we can use the concept of the derivative.

First, we need to find the derivative of f(x). The derivative of f(x) = sqrt(x+1) can be found using the power rule of differentiation.

The derivative of f(x) = sqrt(x+1) is f'(x) = 1 / (2 * sqrt(x+1)).

Next, we substitute x = 0 into the derivative to find the slope of the tangent line at the point (0,1).

f'(0) = 1 / (2 * sqrt(0+1)) = 1 / (2 * sqrt(1)) = 1 / (2 * 1) = 1/2.

So, the slope of the tangent line at (0,1) is 1/2.

Now, we can use the point-slope form of a linear equation to find the equation of the tangent line.

Using the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, we substitute the values.

y - 1 = (1/2)(x - 0).

Simplifying, we get y - 1 = (1/2)x.

Finally, we can rewrite the equation in slope-intercept form, y = mx + b, where b is the y-intercept.

y = (1/2)x + 1.

Therefore, the equation of the tangent line to the graph of f(x) = sqrt(x+1) at the point (0,1) is y = (1/2)x + 1.

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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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