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anastassius [24]
3 years ago
13

If f(x) = 8x2 − x3, find f'(2) and use it to find an equation of the tangent line to the curve y = 8x2 − x3 at the point (2, 24)

.

Mathematics
2 answers:
slega [8]3 years ago
6 0
<span>f(x) = 8x2 − x3
f'(x) = 16x - 3x</span>²

f'(2) = 16(2) - 3(2)²
<span>f'(2) = 20

Using point-slope form:

y - y1 = m(x - x1)
y - 24 = 20 ( x - 2)

Therefore, the equation would be:

y = 20x - 16</span>
Svetlanka [38]3 years ago
6 0

f'(2) = 20

Equation of the tangent line → y = 20x - 16

<h3>Further explanation</h3>

To solve this problem there are several basic principles in Derivatives that need to be recalled, namely:

y = a ~ x^n \Rightarrow \frac{dy}{dx} = a ~ n ~ x^{n-1}

y = \sin x \Rightarrow \frac{dy}{dx} = \cos x

y = \cos x \Rightarrow \frac{dy}{dx} = - \sin x

y = u \times v \Rightarrow \frac{dy}{dx} = u' \times v + u \times v'

y = u \div v  \Rightarrow \frac{dy}{dx} = \frac{u' \times v - u \times v'}{v^2}

y = u^n \Rightarrow \frac{dy}{dx} = n \times u^{n-1} \times u'

\texttt{where u and v are functions in variable x}

\texttt{and u' and v' are derivatives of u and v}

Let us now tackle the problem !

<u>Given:</u>

f(x) = 8x^2 - x^3

f'(x) = 8(2x^{2-1}) - 3x^{3-1}

f'(x) = 16x^1 - 3x^2

f'(x) = 16x - 3x^2

f'(2) = 16(2) - 3(2)^2

f'(2) = 32 - 12

f'(2) = \boxed{20}

<em>Equation of the tangent line to the curve</em>

y - y_1 = m ( x - x_1 )

y - 24 = f'(2) ( x - 2 )

y - 24 = 20 ( x - 2 )

y - 24 = 20x - 40

y = 20x - 40 + 24

\boxed {y = 20x - 16}

<h3>Learn more</h3>
  • Implicit Differentiation : brainly.com/question/4711711
  • Logarithmic Differentiation : brainly.com/question/9226310
  • Calculus Problem : brainly.com/question/11237537

<h3>Answer details</h3>

Grade: High School

Subject: Mathematics

Chapter: Differentiation

Keywords: Maximum , Minimum , Value , Function , Variable , Derivation , Differentiation

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