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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Using the definition of dervtive of a function at point. f(a) = ly fotos) Ee hore cf te imve'ot te tet FOr’ Ei site {wt he ice os Er eturas flz)=32%-10z+11:a=-3 The siopeis| | Note: type "undefined" if you think the slope of the curve is undefined at the given points.

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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Uploaded ImageUsing the definition of dervtive of a function at point. f(a) = ly fotos) Ee hore cf te imve'ot te tet FOr’ Ei site {wt he ice os Er eturas flz)=32%-10z+11:a=-3 The siopeis| | Note: type "undefined" if you think the slope of the curve is undefined at the given points.

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Asked by: sai
Created at: 2025-06-08 04:23:52

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**Step 1: Set up the definition of the derivative at the given point** To find the slope of the curve at \( a = -3 \), use the definition of the derivative: \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] Given \( f(x) = 3x^2 - 10x + 11 \) and \( a = -3 \): Set up the difference quotient: \[ f'(-3) = \lim_{h \to 0} \frac{f(-3+h) - f(-3)}{h} \] **Explanation:** This step substitutes the specific value \( a = -3 \) into the general derivative formula. The difference quotient gives the average rate of change of the function at points close to \( x = -3 \). --- **Step 2: Calculate \( f(-3+h) \) and \( f(-3) \), substitute, and simplify** Calculate \( f(-3) \): \[ f(-3) = 3(-3)^2 - 10(-3) + 11 = 3(9) + 30 + 11 = 27 + 30 + 11 = 68 \] Calculate \( f(-3+h) \): \[ f(-3+h) = 3(-3+h)^2 - 10(-3+h) + 11 \] \[ = 3(9 - 6h + h^2) - 10(-3 + h) + 11 \] \[ = 27 - 18h + 3h^2 + 30 - 10h + 11 \] \[ = (27 + 30 + 11) + (-18h - 10h) + 3h^2 \] \[ = 68 - 28h + 3h^2 \] Substitute into the difference quotient: \[ f'(-3) = \lim_{h \to 0} \frac{[68 - 28h + 3h^2] - 68}{h} \] \[ = \lim_{h \to 0} \frac{-28h + 3h^2}{h} \] \[ = \lim_{h \to 0} \frac{h(-28 + 3h)}{h} \] \[ = \lim_{h \to 0} (-28 + 3h) = -28 \] **Explanation:** All terms are substituted and simplified. The limit is evaluated as \( h \to 0 \), yielding the slope at \( x = -3 \). --- **Final Answer Summary:** The slope of the curve at \( x = -3 \) is **-28**.

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