Question

Assume that \( f(x) \) is continuous and strictly increasing on \( [1,4] \) with \( f(1)=1 \) and \( f(4)=3 \). Let \( g(y):[1,3] \rightarrow[1,4] \) be the inverse of \( f(x) \) on [1,4]. (Note: \( g(y) \) is also continuous and increasing on \( [1,3] \) and as such integrable on \( [1,3] \).) a) [4 marks] By using the geometric interpretation of the integral, for each of the following statements indicate if it is True or False i) \( 3 \leq \int_{1}^{4} f(t) d t \leq 9 \). ii) \( \int_{1}^{4} f(t) d t+\int_{1}^{3} g(t) d t=11 \). b) [4 marks] Let \( F(x)=\int_{1}^{x} f(t) d t \). Assume that \( G(y)=\int_{1}^{y} g(t) d t \). By making use of the first FTC answer the following questions. i) Assume that \( f(2)=2.5 \). What is \( F^{\prime}(2) \) ? ii) Assume that \( f(2)=2.5 \). What is \( G^{\prime}(2.5) \). c) [4 marks] Monte Carlo Method: Assume that a random number generator is used to generate pairs of numbers \( (x, y) \) where \( 1 \leq x \leq 4 \) and \( 0 \leq y \leq 3 \). Suppose that 10000 such pairs are generated and 9025 pairs are such that for the given \( (x, y) \) we have \( y \leq f(x) \). Use this to estimate \( \int_{1}^{4} f(t) d t \) and briefly explain how you came up with your estimate.

          Assume that \( f(x) \) is continuous and strictly increasing on \( [1,4] \) with \( f(1)=1 \) and \( f(4)=3 \). Let \( g(y):[1,3] \rightarrow[1,4] \) be the inverse of \( f(x) \) on [1,4]. (Note: \( g(y) \) is also continuous and increasing on \( [1,3] \) and as such integrable on \( [1,3] \).)
a) [4 marks] By using the geometric interpretation of the integral, for each of the following statements indicate if it is True or False
i) \( 3 \leq \int_{1}^{4} f(t) d t \leq 9 \).
ii) \( \int_{1}^{4} f(t) d t+\int_{1}^{3} g(t) d t=11 \).
b) [4 marks] Let \( F(x)=\int_{1}^{x} f(t) d t \). Assume that \( G(y)=\int_{1}^{y} g(t) d t \). By making use of the first FTC answer the following questions.
i) Assume that \( f(2)=2.5 \). What is \( F^{\prime}(2) \) ?
ii) Assume that \( f(2)=2.5 \). What is \( G^{\prime}(2.5) \).
c) [4 marks] Monte Carlo Method: Assume that a random number generator is used to generate pairs of numbers \( (x, y) \) where \( 1 \leq x \leq 4 \) and \( 0 \leq y \leq 3 \). Suppose that 10000 such pairs are generated and 9025 pairs are such that for the given \( (x, y) \) we have \( y \leq f(x) \). Use this to estimate \( \int_{1}^{4} f(t) d t \) and briefly explain how you came up with your estimate.

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Assume that f(x) is continuous and strictly increasing on [1,4] with f(1)=1 and f(4)=3. Let g(y):[1,3] →[1,4] be the inverse of f(x) on [1,4]. (Note: g(y) is also continuous and increasing on [1,3] and as such integrable on [1,3].)
a) [4 marks] By using the geometric interpretation of the integral, for each of the following statements indicate if it is True or False
i) 3 ≤∫1^4 f(t) d t ≤ 9.
ii) ∫1^4 f(t) d t+∫1^3 g(t) d t=11.
b) [4 marks] Let F(x)=∫1^x f(t) d t. Assume that G(y)=∫1^y g(t) d t. By making use of the first FTC answer the following questions.
i) Assume that f(2)=2.5. What is F^'(2) ?
ii) Assume that f(2)=2.5. What is G^'(2.5).
c) [4 marks] Monte Carlo Method: Assume that a random number generator is used to generate pairs of numbers (x, y) where 1 ≤ x ≤ 4 and 0 ≤ y ≤ 3. Suppose that 10000 such pairs are generated and 9025 pairs are such that for the given (x, y) we have y ≤ f(x). Use this to estimate ∫1^4 f(t) d t and briefly explain how you came up with your estimate.

Added by Jing W.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Suman Saurav Thakur

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### a) i) \( 3 \leq \int_{1}^{4} f(t) d t \leq 9 \) ** Show more…

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