How do I find the linear approximation of the function #g(x)=(1+x)^(1/3)# at #a=0#?
We have
#g(a) = g(0) = (1 +0)^(1/3) = 1#
Now taking the derivative.
#g'(x) = 1/3(1 + x)^(-2/3)#
#g'(a) = g'(0) = 1/3(1 + 0)^(-2/3) = 1/3#
Now we find the equation of the tangent.
#y -y_1 = m(x - x_1)#
#y - 1 = 1/3(x - 0)#
#y = 1/3x + 1#
As you can see this approximates the function relatively well for value of
Hopefully this helps!
By signing up, you agree to our Terms of Service and Privacy Policy
To find the linear approximation of the function ( g(x) = (1+x)^{\frac{1}{3}} ) at ( a = 0 ), follow these steps:
- Find the first derivative of ( g(x) ), denoted ( g'(x) ).
- Evaluate ( g'(0) ) to find the slope of the tangent line at ( x = 0 ).
- Use the point-slope form of a line to write the equation of the tangent line.
Let's go through each step:
- The first derivative of ( g(x) ) is ( g'(x) = \frac{1}{3}(1+x)^{-\frac{2}{3}} ).
- Evaluate ( g'(0) ): ( g'(0) = \frac{1}{3}(1+0)^{-\frac{2}{3}} = \frac{1}{3} ).
- Use the point-slope form of a line with the point ( (0, g(0)) = (0, 1) ) and slope ( g'(0) = \frac{1}{3} ):
[ y - y_1 = m(x - x_1) ] [ y - 1 = \frac{1}{3}(x - 0) ] [ y - 1 = \frac{1}{3}x ]
This is the equation of the tangent line at ( x = 0 ).
By signing up, you agree to our Terms of Service and Privacy Policy
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.
- How do you find a power series representation for #(arctan(x))/(x) # and what is the radius of convergence?
- How do you find the Taylor polynomial for #1/(2-x)#?
- How do you use a Taylor series to prove Euler's formula?
- How do you find a power series representation for #f(x)= 1/(1+x)# and what is the radius of convergence?
- How do you find the radius of convergence #Sigma x^n/3^n# from #n=[0,oo)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7