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Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Given:

Find $f\left ( x \right )$

(x^2-5x+2)^5(36x^3-105x^2+12x)

(x^2-5x+2)^4(36x^3-45x^2+12x)

(15x^2)(x^2-5x+2)^4

(x^2-5x+2)^4(60x^2-150)

(x^2-5x+2)^4(36x^3-105x^2+12x)

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Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Given:

Find $f\left ( x \right )$

Your answer:

(x^2-5x+2)^5(36x^3-105x^2+12x)

Your answer:

(x^2-5x+2)^4(36x^3-45x^2+12x)

Your answer:

(15x^2)(x^2-5x+2)^4

Your answer:

(x^2-5x+2)^4(60x^2-150)

Correct answer:

(x^2-5x+2)^4(36x^3-105x^2+12x)

Explanation:

Computation of the derivative requires the use of the Product Rule and Chain Rule.

The Product Rule is used in a scenario when one has two differentiable functions multiplied by each other:

f(x)*g(x)

$\frac{d}{dx}[f(x)*g(x)]=f(x)*g'(x)+g(x)*f(x)$

This can be easily stated in words as: "First times the derivative of the second, plus the second times the derivative of the first."

In the problem statement, we are given:

f(x)=3x^2(x^2-5x+2)^5

3x^2 is the "First" function, and (x^2-5x+2)^5 is the "Second" function.

The "Second" function requires use of the Chain Rule.

When:

$y=(f(x))^a \rightarrow \frac{dy}{dx}=a*(f(x))^{a-1}*f'(x)$

Applying these formulas results in:

f'(x)=(3x^2)[5(x^2-5x+2)^4(2x-5)]+(x^2-5x+2)^5(6x)

Simplifying the terms inside the brackets results in:

f'(x)=(3x^2)[(10x-25)(x^2-5x+2)^4]+(x^2-5x+2)^5(6x)

We notice that there is a common term that can be factored out in the sets of equations on either side of the "+" sign. Let's factor these out, and make the equation look "cleaner".

f'(x)=(x^2-5x+2)^4[(3x^2)(10x-25)+(6x)(x^2-5x+2)]

Inside the brackets, it is possible to clean up the terms into one expanded function. Let us do this:

f'(x)=(x^2-5x+2)^4[30x^3-75x^2+6x^3-30x^2+12x]

Simplifying this results in one of the answer choices:

f'(x)=(x^2-5x+2)^4(36x^3-105x^2+12x)

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Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

$f(x)=x^2\tan^{-1}(3x^5)$

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

$f(x)=x^2\tan^{-1}(3x^5)$

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Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

Flashcards: Determine Limits Using Algebraic Properties and the Squeeze Theorem

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