Math Problem Statement
Does the graph of the following curve have a tangent at the origin? Give reasons for the answer.
f(x) = { x^2 * sin(1/x), x ≠ 0; 0, x = 0 }
Solution
To determine whether the curve defined by the function has a tangent at the origin, we need to investigate two key things:
- Continuity at : The graph must be continuous at the origin to have a tangent at that point.
- The derivative at : The slope of the tangent line is given by the derivative of the function at the origin.
Step 1: Checking Continuity at
The function is given as:
\begin{cases} x^2 \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ To check continuity at $$x = 0$$, we must see if: $$\lim_{x \to 0} f(x) = f(0)$$ At $$x = 0$$, $$f(0) = 0$$. Now we check the limit: $$\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)$$ Since $$\sin\left(\frac{1}{x}\right)$$ oscillates between -1 and 1, we have: $$- x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2$$ As $$x \to 0$$, both bounds go to 0, so by the Squeeze Theorem, we have: $$\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0$$ Therefore, $$\lim_{x \to 0} f(x) = f(0)$$, confirming that the function is continuous at $$x = 0$$. ### Step 2: Checking the Derivative at $$x = 0$$ To find if there is a tangent, we need to calculate the derivative of $$f(x)$$ at $$x = 0$$. The derivative is given by the limit: $$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}$$ Since $$f(0) = 0$$, we have: $$f'(0) = \lim_{h \to 0} \frac{h^2 \sin\left(\frac{1}{h}\right)}{h}$$ Simplifying the expression: $$f'(0) = \lim_{h \to 0} h \sin\left(\frac{1}{h}\right)$$ Again, since $$\sin\left(\frac{1}{h}\right)$$ oscillates between -1 and 1, we have: $$- h \leq h \sin\left(\frac{1}{h}\right) \leq h$$ As $$h \to 0$$, both bounds approach 0, so by the Squeeze Theorem: $$f'(0) = 0$$ ### Conclusion The function $$f(x)$$ is continuous at $$x = 0$$ and the derivative at $$x = 0$$ exists and equals 0. Thus, the graph has a tangent at the origin with a slope of 0. The tangent line is horizontal. Would you like a more detailed explanation on any part of this process, or have any additional questions? Here are some related questions for further exploration: 1. How do we apply the Squeeze Theorem in more complex limits? 2. What are some other examples of functions with oscillations near 0? 3. How do you differentiate piecewise functions at points of discontinuity? 4. What would happen if the function were not continuous at $$x = 0$$? 5. How does the concept of the derivative relate to physical motion, such as velocity at a point? **Tip:** Always check the continuity of a function before trying to compute its derivative at a point. This ensures you're working with a well-defined function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Squeeze Theorem
Piecewise Functions
Formulas
f'(a) = lim_{h → 0} (f(a + h) - f(a)) / h
Theorems
Squeeze Theorem
Suitable Grade Level
Undergraduate Mathematics or Advanced High School (Grades 11-12)