Calculus I, Fall 2013

Quiz 3

Nov 14, 2013

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1. Prove that Rolle’s Theorem implies the Mean Value Theorem. (Need not prove Rolle’s Theorem)

2. If f(x) is a differentiable function for all x ∈ R and \frac{df}{dx} = (x − 1)(x − 2)^{2}(x − 4). Find all x, if any, where f attains a local minimum or a local maximum.

3. Find lim_{x→∞}(\frac{x + 2}{x − 1})^{x}.

4. A light ray travels from (0, 1), gets reflected at some point (x, 0) on the x-axis and arrives at (3, 2). According to Fermat’s principle, the point of reflection is chosen to minimize the total travel distance (0, 1) − (x, 0) − (3, 2) (since the light speed c is assumed to be a constant on y > 0). Formulate this as an optimization problem, find x and explain why the answer you get is actually a global minimum.

Calculus I, Fall 2013

Quiz 3

Nov 14, 2013

Show all details.

1. Prove that Rolle’s Theorem implies the Mean Value Theorem. (Need not prove Rolle’s Theorem)

2. If f(x) is a differentiable function for all x ∈ R and \frac{df}{dx} = (x − 1)(x − 2)^{2}(x − 4). Find all x, if any, where f attains a local minimum or a local maximum.

3. Find lim_{x→∞}(\frac{x + 2}{x − 1})^{x}.

4. A light ray travels from (0, 1), gets reflected at some point (x, 0) on the x-axis and arrives at (3, 2). According to Fermat’s principle, the point of reflection is chosen to minimize the total travel distance (0, 1) − (x, 0) − (3, 2) (since the light speed c is assumed to be a constant on y > 0). Formulate this as an optimization problem, find x and explain why the answer you get is actually a global minimum.