Precálculo preguntas y respuestas

Given two points for an exponential function, 1. Use the two points to find the growth rate, k . Write an exponential model for each point, then solve this system of two equations for k . 2. Use either point with the k you found to find the initial amount at time zero, \( \mathrm{A}_{\mathrm{o}} \). 3. Doubling time is when the amount is \( 2^{*} \mathrm{~A}_{\mathrm{o}} \). 4. Use the values of k and \( \mathrm{A}_{\mathrm{o}} \) to calculate the amount for a given time or to find the time to reach a specific amount in the future. The count in a bacteria culture was 600 after 20 minutes and 1900 after 30 minutes. Assuming the count grows exponentially. You may enter the exact value or round to 2 decimal places. What was the initial size of the culture? Find the doubling period. Find the population after 105 minutes. When will the population reach 14000 .

A wooden artifact from an ancient tomb contains 40 percent of the carbon-14 that is present in living trees. How long ago, to the nearest year, was the artifact made? (The half-life of carbon-14 is 5730 years.) years

2. Domonstre por indução matemática que, para todo intelro positivo \( n \), é válida a proprosição \( P(n) \) : \( \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} \)

2) \( f(x)=\frac{x^{2}+x}{3 x^{2}-12} \)

Evaluate or simplify the expression without using a calculator. \( e^{\ln 3 x^{3}} \) \( e^{\ln 3 x^{3}}=\square \)

Evaluate or simplify the expression without using a calculator. \( e^{\ln 146} \)

27 Multiple Choice 1 point Solve the problem. The population of an animal species in a certain area is modeled by \( \mathbf{F}(\mathrm{t})=400 \mathrm{log}(2 t+3) \) where \( t \) is the time in months since the species was introduced to the area. Find the population of this species in the area 6 months after the species is introduced. 74 704 240

6. Graph \( y=\log _{\frac{1}{2}}(x)+ \) 2. Analyze the function where \( 0<b<1 \). Domain: Range: \( \lim _{x \rightarrow 0^{+}} f(x) \)

Find the vertical and horizontal asymptotes for \( g(x)=\frac{\sqrt{x^{2}-4}}{2-x} \)

Hence, determine a general formula for the pattern \( 0 ;-6 ;-20 ;-42 \ldots \) Simplify your answer as far as possible. ESTION 4 c) \( =-2 x^{2}+2 \) and \( g(x)=2^{x}+1 \) are the defining equations of graphs \( f \) and \( g \) respectively. Write down an equation for the asymptote of \( g \). Sketch the graphs of \( f \) and \( g \) on the same set of axes, clearly showing ALL intercepts with the axes, turning points and asymptotes. Write down the range of \( f \). Determine the maximum value of \( h \) if \( h(x)=3 /(x) \). What transformation does the graph of \( y=f(x) \) undergo in order to obtain the