Section II Part A — Full Practice Exam 1

Questions in this quiz (2)

1. A community center started tracking attendance at its annual fundraising event. In the initial year, 2005 $$(t=0)$$, the attendance was $$80$$ people. Five years later $$(t=5)$$, the attendance increased to $$210$$ people. The attendance can be modeled by the function $$A(t)=c+d\ln(t+1),$$ where $$A(t)$$ is the number of attendees and $$t$$ is the number of years since 2005.(A)(i) Use the given data to write two equations that can be used to determine the constants $$c$$ and $$d$$.(ii) Solve for $$c$$ and $$d$$, giving decimal approximations where appropriate.(B)(i) Find the average rate of change of attendance from $$t=0$$ to $$t=5$$. Express your answer as a decimal and interpret its meaning in context.(ii) Use the average rate of change to estimate the attendance at $$t=25$$. The event continued for additional years, and the attendance data is shown below:(C)(i) Use a graphing calculator to find a logarithmic regression model for the data. Round coefficients to the nearest thousandth. Explain why a logarithmic model is appropriate.(ii) Use your model to estimate the attendance at $$t=30$$.(iii) Explain why the estimate using the average rate of change is greater than the estimate from the logarithmic model.

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2. Let $$f$$ be a function defined for all real numbers such that $$f$$ is decreasing for $$x$$<$$2$$ and increasing for $$x$$>$$2$$. The table below shows selected values of $$f(x)$$:The function $$g$$ is given by $$g(x)=\frac{3x}{\sqrt{36-x^2}}.$$(A)(i) The function $$h$$ is defined by $$h(x)=(g\circ f)(x)=g(f(x))$$. Find the value of $$h(0)$$ as a decimal approximation, or indicate that it is not defined.(ii) Find all values of $$x$$ for which $$f(x)=4$$, or indicate that there are no such values.(B)(i) Find all values of $$x$$, as decimal approximations, for which $$g(x)=3$$, or indicate that there are no such values.(ii) Determine the end behavior of $$g$$ as $$x$$ approaches its upper bound. Express your answer using limit notation.(C)(i) Determine whether $$f$$ has an inverse function.(ii) Justify your answer to part (i) using the definition of a function and the information provided in the table.

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