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Home Question Bank Math Pre Calculus Archive 2024 November 10

Pre-calculus Questions from Nov 10,2024

Browse the Pre-calculus Q&A Archive for Nov 10,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Find the horizontal asymptote, if any, of the graph of the rational function. \( g(x)=\frac{12 x^{2}}{4 x^{2}+3} \) Find the vertical asymptotes, if any, and the values of \( x \) corresponding to holes, if any, of the graph of the rational function. \( -h(x)=\frac{x+3}{x(x+8)} \) Follow the seven step strategy to graph the following rational function, \( f(x)=\frac{4 x^{2}}{x^{2}-4} \) point(s) posilble Find the horizontal asymptote, if any, of the graph of the rational function. \( g(x)=\frac{24 x^{2}}{6 x^{2}+1} \) A. The horizontal asymptote is B. There is no horizontal asymptote. (Type an equation.) Find the horizontal asymptote of the given functio \( g(x)=\frac{x^{2}+1 x-9}{x-9} \) Approximate the coordinates of each turning point by graphing \( f(x) \) in the standard viewing rectangle. \( f(x)=-7 x^{2}+13 x-5 \) Find the horizontal asymptote of the given function. \( g(x)=\frac{x^{2}+4 x-5}{x-5} \) Solve for \( x \) and leave your answer in \( \ln \) format \( 3 \sinh ^{2} x-2 \cosh x=2 \) 16. Prove by mathematical induction that, for all positive integers \( n \) (a) \( 1+3+5+\ldots+(2 n-1)=n^{2} \) (b) \( \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}=\frac{n}{(n+1)} \) (c) \( 1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left[\frac{1}{2} n(n+1)\right]^{2} \) (d) \( 3+11+\ldots+(8 n-5)=4 n^{2}-n \) (e) \( \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)} \) (f) \( 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4} \) (g) \( 1+2+2^{2}+\ldots+2^{n-1}=2^{n}-1 \) (h) \( 1 \cdot 3+2 \cdot 3^{2}+4 \cdot 3^{3}+\ldots+n \cdot 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4} \) 16. Prove by mathematical induction that, for all positive integers \( n \) (a) \( 1+3+5+\ldots+(2 n-1)=n^{2} \) (b) \( \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}=\frac{n}{(n+1)} \) (c) \( 1^{3}+2^{3}+3^{3}+\ldots+n^{3}=\left[\frac{1}{2} n(n+1)\right]^{2} \) (d) \( 3+11+\ldots+(8 n-5)=4 n^{2}-n \) (e) \( \frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)} \) (f) \( 1 \cdot 2 \cdot 3+2 \cdot 3 \cdot 4+\ldots+n(n+1)(n+2)=\frac{n(n+1)(n+2)(n+3)}{4} \) (g) \( 1+2+2^{2}+\ldots+2^{n-1}=2^{n}-1 \) (h) \( 1 \cdot 3+2 \cdot 3^{2}+4 \cdot 3^{3}+\ldots+n \cdot 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4} \) 17. Prove \( u s i n g \) mathematical induction that for any positive integer \( n \) (a) \( a^{n}-b^{n} \) is divisible by \( a-b \) (b) \( a^{2 n-1}+b^{2 n-1} \) is divisible by \( a+b \) (c) \( (a b)^{n}=a^{n} b^{n} \) for \( a>0 \) (d) \( \left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}} \) (e) \( n^{3}+5 n \) is divisible by 3 (f) \( 5^{n}-1 \) is divisible by 4 (a) \( A^{n}-1 \) is divisible bu 3
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