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Let $U \subset \mathbb{R}^n$ be an open ball around the origin and let $f: U \rightarrow \mathbb{R}$ be a function. For each $t \in [0, 1]$, we define a function $f_t: U \rightarrow \mathbb{R}$ by $f_t(x) := f(tx)$. Prove that the following statements are equivalent: (a) $\lim_{x \rightarrow 0} f(x) = L$. (b) The family of functions $f_t$ converges uniformly to the constant function $L$ in $U$ as $t \rightarrow 0$. Reminder: We say that $f_t \rightarrow L$ uniformly in $U$ if: For every $\varepsilon > 0$, there exists $\delta > 0$ such that if $0 < t < \delta$, then $|f_t(x) - L| < \varepsilon$ for all $x \in U$.

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To avoid dehydration, a toddler should be offered milk or juice at every meal. Group of answer choices true false No answer text provided. No answer text provided.

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$\text{______ is an unrelenting lack of pleasure in life.}$ Anger Fear Anxiety Depression

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Suppose in 2021 the Home country imported $600 million in foreign assets and exported $200 in foreign assets. The rest of the world imported $600 million in Home assets and exported (to Home) $200 million in Home assets. Suppose Home's capital account was zero and valuation effects during the year amount to $50 million in capital gains for Home. a. What was Home's financial account? Explain. b. By how much did Home's net international investment position (net wealth) change during the year? (Hint: Include any capital gains or losses to financial account to get NIIA). c. Which was greater, home's national savings or investment in 2021? Explain.

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4. 3 Ví d? 3: Mua ng?u nhiên m?t t? vé s? có 6 ch? s?. Tính xác su?t? a) Mua ???c m?t t? vé s? có 6 ch? só?, ?i?u là s? l?. b) Mua ???c m?t t? vé s? có 6 ch? s?, ?i?u là s? l?. Bi?t rângg vé s? ?ã mua có các ch? s? $ <7 $. c) Mua ???c môt t? vé s? có 6 ch? só, có t?t c? các ch? s? ?i?u khác 0 . Bi?t râng vé s? ?ã mua có các ch? s? ?i?u là s? ch?n. 4. Quan sát nhûng sinh viên chuyên ngành Toán (SVCNT) tháy râng : $ 70 \% $ thích nghiên c?u, $ 55 \% $ thích v?n th? và $ 35 \% $ thich c? hai. Tính xác suát ?é SVCNT thích v?n th? s? thích nghiên c?u và xác suát ?ê SVCNT thích nghiên c?u s? thích v?n th?. ThSN Nyhin 7

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Question 10 (1 point) Which of the following defines the Internet of Things? A) an application of technology, governed by business logic and structured inputs, aimed at automating business processes B) iterative exploration and investigation of past business performance to gain insight for business planning and decision making C) a system of interrelated computing devices, objects, mechanical and digital machines, people, and animals equipped with unique identifiers that can transfer data and communicate with one another over the Internet D) computing systems that can perform tasks that normally require human intelligence

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Question 2 (25 marks in total) Beth has a choice of two assets. The first is a risk-free asset that offers a rate of return $ r_{f}=5 \% $; the second is a risky asset that has an expected rate of return of $ r_{m}=22 \% $ and a standard deviation of $ \sigma_{m}=35 \% $. a) (10 marks) Suppose that Beth's utility function has the form \[ u\left(r_{x}, \sigma_{x}\right)=r_{x}-3 \sigma_{x} \] Plot Beth's budget line and a few of her indifference curves in a graph. Be sure to label everything in your graph. How much of her portfolio will she invest in the risky asset? What is the expected return and standard deviation of her portfolio? What is her level of utility? b) (15 marks) Suppose that Beth's utility function has the form \[ u\left(r_{x}, \sigma_{x}\right)=\min \left\{r_{x}, 20 \%-2 \sigma_{x}\right\} \] Plot Beth's budget line and a few of her indifference curves in a graph. Be sure to label everything in your graph. How much of her portfolio will she invest in the risky asset? What is the expected return and standard deviation of her portfolio? What is her level of utility?

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At a certain temperature, 3.2x10-4 g of silver phosphate is soluble in 50.0 mL of a solution. What is the pKsp at that temperature?

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Round your final answer to three decimal places, if necessary. Find the angle between the vectors a=-2i+2j-2k and b=3i+5j+3k. heta ~~ radians Round your final answer to three decimal places, if necessary Find the angle between the vectors a=-2i +2j-2k and b=3i +5i+3k O~ radians

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Suppose that Clair was going to receive $50 next year and then the same amount again in 2 years. What is the total present discounted value of that payment stream when the interest rate is 70%? (Do not include a $ sign in your response. Round to the nearest two decimal places.) Answer: Check

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