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Home Question Bank Math Pre Calculus Archive 2025 January 28

Pre-calculus Questions from Jan 28,2025

Browse the Pre-calculus Q&A Archive for Jan 28,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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EJERCICIO \# 1.- Calculo de vectores \( \mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle \) y \( \mathbf{c}=\langle 2,6,9\rangle \). Encuentre el vector o escalar indicado. \( \begin{array}{ll}\text { 1. } \mathbf{a}+(\mathbf{b}+\mathbf{c}) & \text { 3. } \mathbf{b}+2(\mathbf{a}-3 \mathbf{c}) \\ \text { 2. } 2 \mathbf{a}-(\mathbf{b}-\mathbf{c}) & \text { 4. } 4(\mathbf{a}+2 \mathbf{c})-6 \mathbf{b}\end{array} \) 6. (a) How many terms of the geometric sequence \( 64 ; 32 ; 16 ; \ldots \) will add up to \( 127 \frac{1}{2} \) ? (b) How many terms of the geometric sequence \( \frac{1}{27} ; \frac{1}{9} ; \frac{1}{3} ; \ldots \) yield a sum of \( \frac{364}{27} \) ? (c) Determine \( n \) if: \( 3+6+12+\ldots \) (to \( n \) terms \( )=765 \) 7. Determine \( m \) in each of the following arithmetic series: (a) \( \sum_{k=1}^{m}(2)^{8-k}=255 \frac{1}{2} \) (b) \( \sum_{p=1}^{m}(-8)\left(\frac{1}{2}\right)^{p-1}=-15 \frac{3}{4} \) (c) \( \sum_{i=1}^{m} 5(1)^{i-2}=500 \) 8. What is the least value of \( p \) for which the series \( \sum_{k=1}^{p} \frac{1}{16}(2)^{k-2}>31 \) ? 9. Mr Deeds gives his son R1 on his 1st birthday, R2 on his 2nd birthday, R4 on his 3rd birthday, R8 on his 4th birthday and so forth. (a) How much will he give his son on his \( 18^{\text {th }} \) birthday? (b) Find the total amount of money he will have given him by that stage? EXERCISE 7 1. Find the sum of each geometric series (With the appropriate formula): (a) \( \frac{2}{27}+\frac{2}{9}+\frac{2}{3}+\ldots \) to 9 terms (b) \( -64+32-16+\ldots \) to 10 terms (c) \( \log 2+\log 4+\log 16+\ldots \) to 15 terms 2. Calculate the sum of the first 12 terms of the geometric sequence \( \frac{1}{4} ;-\frac{1}{2} ; 1 ; \ldots \) 3. (a) How many terms are there in the series \( -64-32-16-\ldots .-\frac{1}{32} \) (b) What is the sum of the series 4. Evaluate: \( -9-6-4-\ldots .-1 \frac{5}{27} \) 5. (a) How many terms of the geometric sequence \( 64 ; 32 ; 16 ; \ldots \) will add up to \( 127 \frac{1}{2} \) ? (b) How many terms of the geometric sequence \( \frac{1}{27} ; \frac{1}{9} ; \frac{1}{3} ; \ldots \) yield a sum of \( \frac{364}{27} \) ? (c) Determine \( n \) if: \( 3+6+12+\ldots \) (to \( n \) terms) \( =765 \) 6. (a) If \( T_{6}=-243 \) and \( T_{3}=72 \) of a geometric sequence determine: (i) the constant ratio (ii) the sum of the first 10 terms (b) If the constant ratio of a geometric sequence is \( -\frac{1}{2} \). The \( 8^{\text {th }} \) term of the same sequence is \( -\frac{5}{32} \). Determine: (i) the first term (i) the first term (ii) the sum of the first 8 terms. \[ f(x)=-(x+1)^{2}(x-2)^{2} \] Answer the questions regarding the graph of \( f \). hen, use this information to graph the function (a) Choose the end behavior of the graph of \( f \). Choose One Graph all vertical and horizontal asymptotes \[ f(x)=\frac{-10 x+11}{-4 x-10} \] Dada la función \( \mathrm{g}(\mathrm{x})=\mathrm{x}+1 \mathrm{y}(\mathrm{f} \circ \mathrm{g})(\mathrm{x})=\cos \left(\mathrm{x}^{2}-1\right) \), halle \( \mathrm{f}(\mathrm{x}) \). Data la seguente funzione: \( f(x)=\sqrt{(x+1)(2-x)} \ln (3 x+1) \) Idominio è 4. Sketch the graphs of each of the following functions and identify the domain, range, intercepts and asymptotes. a) \( f(x)=\log (3-x)^{2}-1 \) b) \( f(x)=\operatorname{coth} x \) 107. Use the exponential growth model, \( A=A_{0} e^{k t} \), to solve this exercise. In 1980, the elderly U.S. population ( 65 and older) was 25.5 million. By 2010, it had grown to 40.3 million. (Source: U.S. Census Bureau) a. Find an exponential growth function that models the data for 1980 through 2010 . b. By which year, to the nearest year, will the elderly U.S. population reach 80 million? (Section 3.5, Example 1) Find each of the following functions. \[ f(x)=\sqrt{x}, g(x)=\sqrt[3]{5-x} \] (a) \( f \circ g \) \( \square \) State the domain of the function. (Enter your answer using interval notation.) \( \square \) (b) \( g \circ f \) \( \square \) State the domain of the function. (Enter your answer using interval notation.) \( \square \) (c) \( f \circ f \) \( \square \) State the domain of the function. (Enter your answer using interval notation.) \( \square \) (d) \( g \circ g \) \( \square \) State the domain of the function. (Enter your answer using interval notation.) \( \square \)
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