Section II Part A — Full Practice Exam 3

Questions in this quiz (2)

1. An increasing, continuous function $$g(x)$$ has outputs shown in the table:A function $$f(x)$$ is shown graphically.(A) (i) Estimate the value of $$g(f^{-1}(1))$$ if it is defined.(ii) If $$h(x)=\\frac{g(x)}{f(x)}$$, identify any discontinuities for $$x$$>$$0$$. State their type and location.(B) Suppose $$ j(x)=\\frac{1}{4.998}\\ln(x)\\cdot e^{2\\sin(\\sqrt{x})}$$(i) Determine the end behavior of $$j(x)$$ as $$x\\to\\infty$$. Express your answer using limit notation.(ii) Determine the vertical asymptote of $$j(x)$$ as $$x\\to0^+$$ using limit notation.(C) Determine whether $$g(x)$$ is best modeled by a linear, quadratic, exponential, or logarithmic function. Justify your answer using the pattern in the table.

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2. A tank contains a certain number of liters of water. A small leak causes water to drain continuously, and the rate at which the water leaves is proportional to the amount remaining. Measurements show that after $$1$$ minute, $$200$$ liters remain, and after $$3$$ minutes, $$128$$ liters remain. The situation can be modeled by an exponential function of the form $$W(t)=Ae^{kt}$$ where $$W(t)$$ is the amount of water in liters at time $$t$$ minutes.(A) Find an exponential function that models this situation.(B)(i) Find the average rate of change of the amount of water from $$t=0$$ to $$t=4$$.(ii) Interpret the meaning of your answer in context.(C) Will the average rate of change from $$t=4$$ to $$t=t_a,$$ where $$t_a$$>$$4,$$ be greater than or less than the average rate of change from $$t=0$$ to $$t=4$$? Justify your answer.(D) Estimate the time when the tank will have $$50$$ liters remaining.

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