Science:Math Exam Resources/Courses/MATH101/April 2015/Question 01 (a)

MATH101 April 2015

  •  Q1 (a)  •  Q1 (b) (i)  •  Q1 (b) (ii)  •  Q1 (b) (iii)  •  Q1 (c)  •  Q1 (d)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c) (i)  •  Q2 (c) (ii)  •  Q2 (c) (iii)  •  Q2 (c) (iv)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q9 (a)  •  Q9 (b)  •  Q10 (a)  •  Q10 (b)  •  Q11 (a)  •  Q11 (b)  •  

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Question 01 (a)

Only one of the following statements is always true; determine which one is true. (Assume f(x) and g(x) are continuous functions.) A. $\int _{-3}^{-2}f(x)dx=-\int _{3}^{2}f(x)dx$. B. $\int _{0}^{1}f(x)g(x)dx=\int _{0}^{1}f(x)dx\cdot \int _{0}^{1}g(x)dx.$ C. If $\int _{1}^{\infty }f(x)dx$ converges and $g(x)\geq f(x)\geq 0$ for all $x$, then $\int _{1}^{\infty }g(x)dx$ converges. D. If $\sum _{n=1}^{\infty }(-1)^{n+1}b_{n}$ converges, then $\sum _{n=1}^{\infty }b_{n}$ also converges. E, When $f(x)$ is positive and concave up, any Trapezoid Rule approximation for $\int _{a}^{b}f(x)dx$ will be an upper estimate for $\int _{a}^{b}f(x)dx$,

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Plug some functions into $f(x)$ and $g(x)$ randomly and guess whether the statement is true or not.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. A. $\int _{-3}^{-2}f(x)dx=-\int _{3}^{2}f(x)dx$. False. Try $f(x)=2x$. Then, $\int _{-3}^{-2}f(x)dx=x^{2}{\bigg |}_{-3}^{-2}=4-9=-5$. On the other hand, $-\int _{3}^{2}f(x)dx=\int _{2}^{3}f(x)dx=x^{2}{\bigg |}_{2}^{3}=9-4=5.$. Indeed, this would be true if $f$ is an even function. B. $\int _{0}^{1}f(x)g(x)dx=\int _{0}^{1}f(x)dx\cdot \int _{0}^{1}g(x)dx.$ False. Try $f(x)=g(x)=x$. Then, $\int _{0}^{1}f(x)g(x)dx=\int _{0}^{1}x^{2}dx={\frac {1}{3}}$. However, $\int _{0}^{1}f(x)dx\cdot \int _{0}^{1}g(x)dx=\left(\int _{0}^{1}xdx\right)^{2}={\frac {1}{4}}$. C. If $\int _{1}^{\infty }f(x)dx$ converges and $g(x)\geq f(x)\geq 0$ for all $x$, then $\int _{1}^{\infty }g(x)dx$ converges. False. Consider $f(x)=e^{-x}$ and $g(x)=1$. Apparently, $0\leq e^{-x}\leq 1$ and $\int _{1}^{\infty }e^{-x}dx=-e^{-x}{\bigg |}_{1}^{\infty }=e^{-1}<\infty$. However, $\int _{1}^{\infty }1dx=\infty$. i.e, it doesn't converge. This would be true if the order of $f(x)$ and $g(x)$ is changed in the inequality: $f(x)\geq g(x)\geq 0$. D. If $\sum _{n=1}^{\infty }(-1)^{n+1}b_{n}$ converges, then $\sum _{n=1}^{\infty }b_{n}$ also converges. False. If $b_{n}={\frac {1}{n}}$, then by the alternating convergence test, we have $\sum _{n=1}^{\infty }(-1)^{n+1}b_{n}<+\infty$. On the other hand, p-series test implies that $\sum _{n=1}^{\infty }b_{n}$ diverges. E, When $f(x)$ is positive and concave up, any Trapezoid Rule approximation for $\int _{a}^{b}f(x)dx$ will be an upper estimate for $\int _{a}^{b}f(x)dx$, True! Since $f(x)$ is positive and concave up, it satisfies $f(\theta x+(1-\theta )y)\leq \theta f(x)+(1-\theta )f(y)$ for $0<\theta <1$ and any $x<y$ on the real line. This means that the graph of $f$ on any interval $[x,y]$ never lies above of the line passing through $(x,f(x))$ and $(y,f(y))$. From this observation and the definition of Trapezoid Rule approximation, we can easily see the statement. Answer: $\color {blue}E$.