Find Left-hand Derivative of a Continuous Piecewise Function at x = 1

Math Problem Statement

Diketahui fungsi yang kontinyu di 1. Tentukan nilai dari turunan kiri fungsi f(x)?

Solution

We are given a piecewise function:

2x, & \text{for } x \geq 1 \\ x^2 + 1, & \text{for } x < 1 \end{cases}$$ The function is continuous at $$x = 1$$, and we are asked to find the left-hand derivative of $$f(x)$$ at $$x = 1$$. ### Steps to Find the Left-hand Derivative The left-hand derivative at $$x = 1$$ is defined as: $$f'_-(1) = \lim_{x \to 1^-} \frac{f(x) - f(1)}{x - 1}$$ Here, we are only concerned with the part of the function where $$x < 1$$, which is $$f(x) = x^2 + 1$$. First, we compute the value of the function at $$x = 1$$ to ensure continuity. #### Continuity Check For continuity at $$x = 1$$, both sides must equal at $$x = 1$$: - From the left (for $$x < 1$$): $$f(1) = 1^2 + 1 = 2$$ - From the right (for $$x \geq 1$$): $$f(1) = 2(1) = 2$$ Since both give $$f(1) = 2$$, the function is continuous at $$x = 1$$. #### Left-hand Derivative Now, calculate the left-hand derivative of $$f(x)$$ for $$x < 1$$, using $$f(x) = x^2 + 1$$: $$f'_-(x) = \frac{d}{dx}(x^2 + 1) = 2x$$ Thus, the left-hand derivative at $$x = 1$$ is: $$f'_-(1) = 2(1) = 2$$ ### Conclusion The left-hand derivative of the function at $$x = 1$$ is $$2$$. --- Do you have any questions or need further details? Here are some related questions for deeper understanding: 1. How do we check for the continuity of a piecewise function at a given point? 2. What is the geometric meaning of the left-hand derivative? 3. How would we calculate the right-hand derivative for this function? 4. Why is it important for a function to be continuous at a point before computing derivatives? 5. How does the limit definition of a derivative help in understanding this problem? **Tip**: When working with piecewise functions, always check both continuity and differentiability at the boundary points.

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Math Problem Analysis

Mathematical Concepts

Continuity
Piecewise Functions
Derivatives
Limits

Formulas

f'_-(x) = lim_{x → 1^-} (f(x) - f(1)) / (x - 1)
f(x) = { 2x, x ≥ 1; x^2 + 1, x < 1 }

Theorems

Definition of Continuity
Limit Definition of Derivatives

Suitable Grade Level

Grades 10-12

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