math104-s21:hw11
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| $$ tf(x) + (1-t) f(y) \geq f(t x + (1-t) y).$$ | $$ tf(x) + (1-t) f(y) \geq f(t x + (1-t) y).$$ | ||
| Prove that $f' | Prove that $f' | ||
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| - | picture from wikipedia | ||
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| 5. (Free discussion problem, no points taken). Let $f: [0,1] \to \R$ be a real bounded function. Assume $f(x) \geq 0$ and $f$ is Riemann integrable. Here is a candidate that measures the area under the graph of $f$: | 5. (Free discussion problem, no points taken). Let $f: [0,1] \to \R$ be a real bounded function. Assume $f(x) \geq 0$ and $f$ is Riemann integrable. Here is a candidate that measures the area under the graph of $f$: | ||
| - | * For each positive integer $n$, let $N(n)$ be the number of points $(x,y) \in \R^2$, such that $x, y \in \Z / 2^n$, and $x \in [0,1], 0 \leq y \leq f(x)$. Namely, we count how many points in the lattice | + | * For each positive integer $n$, let $N(n)$ be the number of points $(x,y) \in \R^2$, such that $x, y \in \Z / 2^n$, and $x \in [0,1], 0 \leq y \leq f(x)$. Namely, we count how many points in the 2d mesh with spacing |
| * Let $A(n) = 2^{-2n} N(n)$. Since the area of a $2^{-n} \times 2^{-n}$ square is $2^{-2n}$. | * Let $A(n) = 2^{-2n} N(n)$. Since the area of a $2^{-n} \times 2^{-n}$ square is $2^{-2n}$. | ||
| - | * Is it true that $\lim_{n \to \infty} A(n) = \int_0^1 f(x)dx$? | + | Is it true that |
| + | $$ \lim_{n \to \infty} A(n) = \int_0^1 f(x)dx ? $$ | ||
| + | To get started, you may assume that $f$ is continuous (we will see next week that continuous functions are Riemann integrable). | ||
math104-s21/hw11.1618637284.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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