Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve

Answer 1

Answer:

$\displaystyle f'(4) = 63$

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

Distributive Property

Algebra I

• Expand by FOIL (First Outside Inside Last)
• Factoring
• Function Notation
• Terms/Coefficients

Calculus

Derivatives

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

Limit Definition of a Derivative: $\displaystyle f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$  

Step-by-step explanation:

Step 1: Define

f(x) = 7x² + 7x + 3

Slope of tangent line at x = 4

Step 2: Differentiate

1. Substitute in function [Limit Definition of a Derivative]:                              $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7(x + h)^2 + 7(x + h) + 3]-(7x^2 + 7x + 3)}{h}$
2. [Limit - Fraction] Expand [FOIL]:                                                                    $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7(x^2 + 2xh + h^2) + 7(x + h) + 3]-(7x^2 + 7x + 3)}{h}$
3. [Limit - Fraction] Distribute:                                                                            $\displaystyle f'(x)= \lim_{h \to 0} \frac{[7x^2 + 14xh + 7h^2 + 7x + 7h + 3] - 7x^2 - 7x - 3}{h}$
4. [Limit - Fraction] Combine like terms (x²):                                                     $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7x + 7h + 3 - 7x - 3}{h}$
5. [Limit - Fraction] Combine like terms (x):                                                      $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7h + 3 - 3}{h}$
6. [Limit - Fraction] Combine like terms:                                                           $\displaystyle f'(x)= \lim_{h \to 0} \frac{14xh + 7h^2 + 7h}{h}$
7. [Limit - Fraction] Factor:                                                                                 $\displaystyle f'(x)= \lim_{h \to 0} \frac{h(14x + 7h + 7)}{h}$
8. [Limit - Fraction] Simplify:                                                                               $\displaystyle f'(x)= \lim_{h \to 0} 14x + 7h + 7$
9. [Limit] Evaluate:                                                                                                 $\displaystyle f'(x) = 14x + 7$

Step 3: Find Slope

1. Substitute in x:                                                                                                $\displaystyle f'(4) = 14(4) + 7$
2. Multiply:                                                                                                           $\displaystyle f'(4) = 56 + 7$
3. Add:                                                                                                                  $\displaystyle f'(4) = 63$

This means that the slope of the tangent line at x = 4 is equal to 63.

Hope this helps!

Topic: Calculus AB/1

Unit: Chapter 2 - Definition of a Derivative

(College Calculus 10e)