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

Answers: 2

Show answers

Answers

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

yes because there is exactly one y value for every x value

for function there can't be more than one value for every x value

Answer from: Quest

step-by-step explanation:

Another question on Mathematics

Mathematics, 20.06.2019 18:04

Write a positive or negative number to represent each situation. a. you win 150 dollars in a contest b. a loss of 15 yards in a football game c. 75 feet below sea level d. 35 degrees above 0 f e. a debt of 24.75 dollars

Answers: 1

Answer

Mathematics, 21.06.2019 23:30

Choose the correct translation for the following statement. it is at most ten. x< 10 x< 10 x> 10 x> 10

Answers: 1

Answer

Mathematics, 21.06.2019 23:40

Type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar.a city water tank holds 20 gallons of water. a technician empties 25% of the tank. how many more gallons of water must be removed from thetank so that it has 5 of the water that it started with: the technician must removemore gallons of water for the tank to have 5 of the water that it started with.

Answers: 1

Answer

Mathematics, 22.06.2019 02:00

Currently, the largest cruise ship has a tonnage of 225,282 gross tons (gt). which of the following is a reasonable estimate for the tonnage of this cruise ship? a. 2 × 10^5 gt b. 2 × 10^6 gt c. 2 × 10^-5 gt d. 2 × 10^-6 gt

Answers: 1

Answer

You know the right answer?

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

Questions

Mathematics, 22.01.2021 20:20

As an archer, Oliver has a shooting percentage of 85%. That means he hits the bulls eye 85% of the shots he attempts. If he attempts 410 shots this se...

Mathematics, 22.01.2021 20:20

How to make a simplified expression for 4(y + 3) - 5...

Mathematics, 22.01.2021 20:20

Tell me a please is a time limit on this ...

Mathematics, 22.01.2021 20:20