Use the limit definition of the derivative to find the slope of the tangent line to the curve
f(x)= 7x^2 + 7x + 3 at x= 4

Answer:

[tex]\displaystyle f'(4) = 63[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Distributive Property

Algebra I

Expand by FOIL (First Outside Inside Last)FactoringFunction NotationTerms/Coefficients

Calculus

Derivatives

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

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

Step-by-step explanation:

Step 1: Define

f(x) = 7x² + 7x + 3

Slope of tangent line at x = 4

Step 2: Differentiate

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

Step 3: Find Slope

Substitute in x:                                                                                                [tex]\displaystyle f'(4) = 14(4) + 7[/tex]Multiply:                                                                                                           [tex]\displaystyle f'(4) = 56 + 7[/tex]Add:                                                                                                                  [tex]\displaystyle f'(4) = 63[/tex]

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)

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