Find the tangent line approximation for sqrt(3+x) near x=0.
when x = 0, y = sqrt 3
y = (x+3)^.5
so
dy/dx = .5 (x+3)^-.5
slope of tangent line at x = 0 is thus
.5/sqrt 3
y = m x + b
sqrt 3 = (.5/sqrt 3)(0) + b
b = sqrt 3
y = (.5/sqrt 3) x + sqrt 3
Thank you
Well, let's find the tangent line approximation for sqrt(3+x) near x=0. So, the derivative of sqrt(3+x) is 1/2sqrt(3+x). Plugging in x=0, we get 1/2sqrt(3+0)=1/2sqrt(3)=sqrt(3)/2. So, the equation of the tangent line is y=sqrt(3)/2*x+sqrt(3)/2. But beware of this tangential relationship, it might just be yanking your chain!
To find the tangent line approximation for √(3+x) near x=0, you can follow the steps outlined below:
Step 1: Calculate the derivative of the function √(3+x) with respect to x. This will give you the slope of the tangent line at any point on the curve.
Let's start by applying the power rule for derivatives:
d/dx (√(3+x)) = 1/(2√(3+x)) * d/dx (3+x)
Notice that the derivative of 3+x with respect to x is 1. Therefore, simplifying the expression further, we have:
d/dx (√(3+x)) = 1/(2√(3+x))
Step 2: Evaluate the derivative at the given value x=0. We want to find the slope of the tangent line at x=0, so substitute x=0 into the derivative expression:
d/dx (√(3+x)) = 1/(2√(3+0)) = 1/(2√3) = √3/6
Step 3: Use the point-slope form of a line to write the equation of the tangent line. We know that the slope of the tangent line at x=0 is √3/6. We also know that the point on the tangent line is (0, √(3+0)) = (0, √3).
The point-slope form of a line is given by:
y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Substituting the values, we have:
y - √3 = (√3/6)(x - 0)
Simplifying, we get:
y - √3 = √3/6 * x
Step 4: Optionally, simplify and rewrite the equation of the tangent line. Multiply both sides of the equation by 6 to remove the fraction:
6y - 6√3 = √3x
Finally, rearrange the equation to put it in slope-intercept form:
√3x - 6y = -6√3
or
y = (√3/6)x - √3
Now you have the equation of the tangent line approximation for √(3+x) near x=0.