Question

Someone, please help with this Calculus problem!!! ASAP

Answer 1

Answer:

(C) 13/6

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

Point-Slope Form: y - y₁ = m(x - x₁)  

• x₁ - x coordinate
• y₁ - y coordinate
• m - slope

Function Notation

Exponential Properties: $\sqrt[n]{x} = x^\frac{1}{n}$

Calculus

The definition of a derivative is the slope of the tangent line.

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

Step 1: Define

f(x) = ∛x

Tangent Line Point (8, 2)

Find approximation of f(10)

Step 2: Differentiate

1. Rewrite Function:                                    $f(x) = x^\frac{1}{3}$
2. Differentiate [Basic Power]:                    $f'(x) = \frac{1}{3} x^{\frac{1}{3} -1}$
3. Simplify Derivative:                                 $f'(x) = \frac{1}{3} x^{\frac{-2}{3}}$
4. Rewrite Derivative:                                  $f'(x) = \frac{1}{3x^{\frac{2}{3} }}$

Step 3: Find Equation of Tangent Line

Tangent Point (8, 2)

Find instantaneous slope

1. Substitute in x:                    $f'(8) = \frac{1}{3(8)^{\frac{2}{3} }}$
2. Exponents:                          $f'(8) = \frac{1}{3(4)}$
3. Multiply:                              $f'(8) = \frac{1}{12}$

This is our slope of the tangent line at (8, 2)

Find instantaneous equation

1. Substitute [PSF]:                    $y - 2 = \frac{1}{12} (x-8)$

Step 4: Find Approximation

Evaluation f(10)

1. Substitute in x:                    $y - 2 = \frac{1}{12} (10-8)$
2. Subtract:                              $y - 2 = \frac{1}{12} (2)$
3. Multiply:                               $y - 2 = \frac{1}{6}$
4. Isolate y:                              $y = \frac{1}{6} + 2$
5. Add:                                     $y = \frac{13}{6}$

Here we see that the approximation would be 13/6 using the tangent line approximation (calculus). Therefore, C is the correct answer.