Mathematics, 09.02.2021 09:00 rnunez3

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

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Answer from: sadieruegner393

$\displaystyle f'(4) = 63$

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: $\displaystyle f'(x)= \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

Step-by-step explanation:

Step 1: Define

f(x) = 7x² + 7x + 3

Slope of tangent line at x = 4

Step 2: Differentiate

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

Step 3: Find Slope

Substitute in x: $\displaystyle f'(4) = 14(4) + 7$ Multiply: $\displaystyle f'(4) = 56 + 7$ 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)

Answer from: Quest

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Answer from: Quest

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Use the limit definition of the derivative to find the slope of the tangent line to the curve f(x)=...

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