# How do you use a linear approximation or differentials to estimate #(8.06)^(⅔)#?

See below

By signing up, you agree to our Terms of Service and Privacy Policy

To estimate ( (8.06)^{\frac{2}{3}} ) using linear approximation or differentials, follow these steps:

- Choose a point close to ( 8.06 ) to use as the base for the linear approximation. Let's choose ( a = 8 ), which is close to ( 8.06 ).
- Find the derivative of the function ( f(x) = x^{\frac{2}{3}} ).
- Use the linear approximation formula ( L(x) = f(a) + f'(a)(x - a) ), where ( L(x) ) is the linear approximation, ( f(a) ) is the value of the function at ( x = a ), and ( f'(a) ) is the derivative of the function evaluated at ( x = a ).
- Substitute ( x = 8.06 ) into the linear approximation formula to estimate ( (8.06)^{\frac{2}{3}} ).

Let's go through the steps:

- Choose ( a = 8 ).
- ( f(x) = x^{\frac{2}{3}} ) implies ( f'(x) = \frac{2}{3}x^{-\frac{1}{3}} ).
- At ( x = 8 ), ( f'(8) = \frac{2}{3} \times 8^{-\frac{1}{3}} = \frac{1}{6} ).
- Linear approximation formula: [ L(x) = f(a) + f'(a)(x - a) ] [ L(x) = (8)^{\frac{2}{3}} + \frac{1}{6}(x - 8) ]
- Estimate ( (8.06)^{\frac{2}{3}} ): [ L(8.06) = (8)^{\frac{2}{3}} + \frac{1}{6}(8.06 - 8) ]

Calculate ( (8)^{\frac{2}{3}} ): [ (8)^{\frac{2}{3}} = 4 ]

Substitute: [ L(8.06) = 4 + \frac{1}{6}(8.06 - 8) ] [ L(8.06) = 4 + \frac{1}{6}(0.06) ] [ L(8.06) = 4 + 0.01 ] [ L(8.06) = 4.01 ]

So, ( (8.06)^{\frac{2}{3}} ) is approximately ( 4.01 ).

By signing up, you agree to our Terms of Service and Privacy Policy

- The sun is shining and a spherical snowball of volume 340 ft3 is melting at a rate of 17 cubic feet per hour. As it melts, it remains spherical. At what rate is the radius changing after 7 hours?
- How do you minimize and maximize #f(x,y)=(2x-3y)^2-1/x^2# constrained to #1<yx^2+xy^2<3#?
- Functions 11 Word Problem?
- How do you find y'' by implicit differentiation for #4x^3 + 3y^3 = 6#?
- How do you use linear approximation to estimate #g(2.95)# and #g(3.05)# if you know that #g(3)=-5#?

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