11.4 exercise
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q1
answer the following.
a. if $y = f(x)$, write down a symbol for the derivative of this function and state its limit definition.
b. if the point where $x = a$ lies on a function $y = f(x)$, identify the feature of the function that $f(a)$ measures.

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Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Derivative Notation", "Limit Definition of Derivative", "Geometric Meaning of Derivative" ], "current_concepts": [ "Derivative Notation", "Limit Definition of Derivative", "Geometric Meaning of Derivative" ] } ## Write down the derivative symbol and its limit definition $$ \begin{aligned} &\text{Symbol: } f'(x) \quad \text{or} \quad \frac{dy}{dx}\ &\text{Limit Definition: } f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \end{aligned} $$ ## Identify the geometric feature measured by the derivative at a point $$ \begin{aligned} &f'(a) = \text{gradient of the tangent line to the curve } y = f(x) \text{ at } x = a \end{aligned} $$ ## Question a Symbol: $f'(x)$ (or $\frac{dy}{dx}$) Limit definition: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ ## Question b $f'(a)$ measures the gradient (or slope) of the tangent line to the curve $y = f(x)$ at the point where $x = a$. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Calculus", "Limit Definition of Derivative" ] }

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Question

Explanation:

Write down the derivative symbol and its limit definition

Identify the geometric feature measured by the derivative at a point

Answer:

Question a

Question b