let $y = \tan(4x + 7)$.
find the differential $dy$ when $x = 4$ and $dx = 0.1$.
find the differential $dy$ when $x = 4$ and $dx = 0.2$.

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{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Differentials" ], "new_concepts": [], "current_concepts": [ "Differentials" ] } ## Find the general formula for the differential $dy$ $$ \begin{aligned} y &= \tan(4x + 7) \ \frac{dy}{dx} &= \sec^2(4x + 7) \cdot \frac{d}{dx}(4x + 7) \ \frac{dy}{dx} &= 4\sec^2(4x + 7) \ dy &= 4\sec^2(4x + 7) \, dx \end{aligned} $$ ## Evaluate $dy$ when $x = 4$ and $dx = 0.1$ $$ \begin{aligned} 4x + 7 &= 4(4) + 7 = 23 \ \sec^2(23) &= \frac{1}{\cos^2(23)} \approx \frac{1}{(-0.532833)^2} \approx 3.52221 \ dy &= 4 \cdot 3.52221 \cdot 0.1 \approx 1.40888 \end{aligned} $$ ## Evaluate $dy$ when $x = 4$ and $dx = 0.2$ $$ \begin{aligned} dy &= 4 \cdot 3.52221 \cdot 0.2 \approx 2.81777 \end{aligned} $$ ## Question 1 Let $y = \tan(4x + 7)$. Find the differential $dy$ when $x = 4$ and $dx = 0.1$ 1.40888 ## Question 2 Find the differential $dy$ when $x = 4$ and $dx = 0.2$ 2.81777 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Differentials" ] }

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Explanation:

Find the general formula for the differential \(dy\)

Evaluate \(dy\) when \(x = 4\) and \(dx = 0.1\)

Evaluate \(dy\) when \(x = 4\) and \(dx = 0.2\)

Answer:

Question 1

Question 2