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Home Question Bank Math Calculus Archive 2025 January 12

Calculus Questions from Jan 12,2025

Browse the Calculus Q&A Archive for Jan 12,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Siano \( f(x)=\tan \left(x^{2}+x^{3}\right) \) e \( g(x)=5 \sin ^{2} x \). È vero che per \( x \rightarrow 0 \) Scegli un alternativa: \( \quad 5 f-g \) è un infinitesimo di ordine 2 rispetto a \( x \) \( \quad f \) è un infinitesimo di ordine superiore rispetto a \( g \) \( \quad f \cdot g \) è un infinitesimo di ordine 6 rispetto a \( x x \) \( f \sim g \) \( \quad \lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=\frac{1}{5} \) Derivar: \( x^{2}+x y=2 \) a. \( \frac{y}{x}-2 \) b. \( -\frac{y}{x}+2 \) c. \( -\frac{2 y}{x-3}-2 \) d. \( -\frac{y}{x}-2 \) 3. Sand is being dropped at the rate of \( 10 \mathrm{~m}^{3} / \mathrm{min} \) onto a conical pile. If the height of the pile is always twice the base radius, at what rate is the height increasing when the pile is 8 m high? Determine the volume generated by revolving the area between \( y = \sqrt{x} \) and the x-axis from \( x = 0 \) to \( x = 4 \) about the y-axis with the washer method. Calcule chacune des limites suivantes \( \lim _{x \rightarrow 1} \frac{x^{1}-1}{x-1} ; \lim _{x \rightarrow 2} \frac{-x^{2}+3 x-2}{(x-2)^{2}} ; \) \( \lim _{x \rightarrow 1}\left(\frac{1}{1-x}-\frac{2}{1-x^{2}}\right) \) \( \begin{array}{ll}\text { 1. } \lim _{x \rightarrow 2} 7= & \text { 2. } \lim _{x \rightarrow \frac{1}{2}}(4 x+3)= \\ \text { 3. } \lim _{x \rightarrow+}(7-2 x)= & \text { 4. } \lim _{x \rightarrow 0} \frac{x^{2}-9}{x-3}= \\ \text { 5. } \lim _{x \rightarrow 4}\left(2 x^{2}-4 x+5\right)= & \text { 6. } \lim _{x \rightarrow \infty} \frac{3-\sqrt{x}}{9-x}=\end{array} \) 55. Prove that \( \left|\begin{array}{lll}1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2}\end{array}\right|=(y-x)(z-x)(z-y) \) 56. Prove that \( \left|\begin{array}{ccc}x^{2} & 2 x y & y^{2} \\ y^{2} & x^{2} & 2 x y \\ z^{2} & y^{2} & x^{2}\end{array}\right|=\left(x^{3}+y^{3}\right)^{2} \) 57. If \( A=\left(\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right) \) verify that \( A^{2}-4 A-5 I=0 \) 58. Express \( \frac{7+x}{1+x+x^{2}+x^{3}} \), into partial fractions. Assuming that \( -1<x<1 \), obtain an expression for \( \frac{7+x}{1+x+x^{2}+x^{3}} \), give your answer in the form of \( a+b x+c x^{2}+d x^{3}+\cdots \) hence, find the value of the coefficients as far as the term in \( x^{5} \) inclusive. 59. Expand \( (x-2)^{0.5} \) as a series of descending power of x as far as the third term, use expansion to find value of \( \sqrt{2} \) by substitution \( x=100 \) 60. Find the coefficient of the term involving \( x^{10} \) in the expansion of \( \left(x^{6}+\frac{1}{x^{4}}\right)^{15} \) 61. Find the middle term(s) of the following expression \( (x y+1)^{7} \) 62. If the constant term in the expansion of \( \left(\sqrt{x}-\frac{p}{x^{2}}\right)^{10} \) is 405 ,. Find the value of \( P \). 63. Obtain the first four terms of the expansion \( \left(1+\frac{x}{2}\right)^{10} \) in ascending power of \( x \), hence find the value of \( (1.005)^{10} \) correct to four decimal places. 64. Prove that if x is very small that its cube and high power can be neglected \( \sqrt{\frac{1+x}{1-x}}=1+x+\frac{x^{2}}{2} \) by taking \( x=\frac{1}{9} \). Prove that \( \sqrt{5} \) is approximately to \( \frac{181}{81} \) 65. Show that \( a b+5(a-b)=1 \), if \( a=\log _{12} 18 \) and \( b=\log _{24} 54 \) 66. When a polynomial \( p x^{2}+q x+r \) is divided by \( x+1, x-1 \) and \( x-2 \) has a remainder 2,8 and 2 respectively. Evaluate the value of \( p, q \) and \( r \) 67. Find the value of \( x \) and \( y \) in the following system of equations \[ \log x-3 \log y=1, x y=1600 \] 68. Use binomial expansion to evaluate each of the following (a) \( \sqrt{1.01} \) (b) \( \sqrt[3]{0.98} \) 69. Prove that \( \log \left(\frac{m}{n}\right)+\log \left(\frac{n}{z}\right)+\log \left(\frac{z}{m}\right)=0 \) 70. Show that \( 3 \sum_{k=1}^{n} k^{2}+3 \sum_{k=1}^{n} k+\sum_{k=1}^{n} 1=(n+1)^{3}-1 \) 71. Prove that \( \frac{1}{3}+\frac{1}{8}+\frac{1}{15}+\cdots+\frac{1}{n(n+2)}=\frac{n(3 n+5)}{4(n+1)(n+2)} \) 72. In the binomial expansion of \( (1+x)^{n} \), where \( n \) is a positive integer, the coefficient of \( x^{4} \) is \( \frac{3}{2} \) times the sum of the coefficients of \( x^{2} \) and \( x^{3} \). Find the value of \( n \) and determine these three coefficient 73. Write the first four terms of the expansion of \( (1-x)^{-2} \) in ascending powers of \( x \), hence deduce that \( \sum_{n=0}^{\infty}(a+b n) x^{n}=\frac{a+(b-a) x}{(1-x)^{2}} \) Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. \[ \sum_{n=1}^{\infty} \frac{3}{(n+4)(n+5)} \] Ine tourth partial sum is \( s_{4}=U .266 \) I (Type an integer or decimal rounded to four decimal places as needed.) The fifth partial sum is \( \mathrm{S}_{5}=0.3 \). (Type an integer or decimal rounded to four decimal places as needed.) The series appears to convergent To what value does this series converge? A. \( \frac{3}{10} \) B. \( \frac{9}{5} \) C. \( \frac{3}{5} \) D. \( \frac{18}{5} \) E. The series diverges. \begin{tabular}{l} Find the first five partial sums of the given series and determine whether the series appears to be \\ convergent or divergent. If it is convergent, find its approximate sum. \\ The first partial sum is \( \mathrm{S}_{1}=0.5 \). \\ (Type an integer or decimal rounded to four decimal places as needed.) \\ The second partial sum is \( \mathrm{S}_{2}=0.75 \). \\ (Type an integer or decimal rounded to four decimal places as needed.) \\ The third partial sum is \( \mathrm{S}_{3}=0.9000 \). \\ (Type an integer or decimal rounded to four decimal places as needed.) \\ The fourth partial sum is \( \mathrm{S}_{4}=\square \). \\ (Type an integer or decimal rounded to four decimal places as needed.) \\ \( (n+1)(\mathrm{n}+2) \) \\ \hline\end{tabular} Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum. \[ \sum_{n=1}^{\infty} \frac{3}{(n+1)(n+2)} \] The first partial sum is \( \mathrm{S}_{1}=0.5 \). (Type an integer or decimal rounded to four decimal places as needed.) The second partial sum is \( \mathrm{S}_{2}=\square \). (Type an integer or decimal rounded to four decimal places as needed.)
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