Mini Exam 3
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The table gives values for a polynomial function $$f$$ at selected values of $$x$$. What is the average rate of change of $$f$$ over the interval $$[2,5]$$?⌄
Use the average rate of change formula.$$\frac{f(5)-f(2)}{5-2}$$From the table.$$f(5)=9$$ $$f(2)=3$$Substitute.$$\frac{9-3}{5-2}$$$$\frac{6}{3}$$ $$2$$Final answer: $$2$$
A set of data is displayed on a semi-log plot where the vertical axis is logarithmic. The data points form a straight line with positive slope. Which type of function best models the data?⌄
A straight line on a semi-log plot indicates an exponential model. A positive slope means the function is increasing. So, the function represents exponential growth. Final answer: $$\text{B}$$
The function C models cost in dollars and is given by $$C(x)=\frac{800+bx}{x}$$. It is known that $$C(10)=90$$ and $$C(20)=50$$. What is the average rate of change of $$C$$ from $$x=40$$ to $$x=50$$?⌄
Rewrite the function.$$C(x)=\frac{800}{x}+b$$Use $$C(10)=90$$.$$\frac{800}{10}+b=90$$$$80+b=90$$ $$b=10$$So,$$C(x)=\frac{800}{x}+10$$Find values.$$C(40)=\frac{800}{40}+10$$$$20+10=30$$$$C(50)=\frac{800}{50}+10$$$$16+10=26$$Average rate of change.$$\frac{26-30}{50-40}$$$$\frac{-4}{10}$$ $$-\frac{2}{5}$$Final answer: $$-\frac{2}{5}$$
The polar function is given by $$r=2+3\cos\theta$$, for $$0 \le \theta \le 2\pi$$. On which interval(s) is the distance from the origin decreasing?⌄
The distance from the origin decreases when $$r$$ decreases.Differentiate $$r$$ with respect to $$\theta.$$$$\frac{dr}{d\theta}=-3\sin\theta.$$The distance is decreasing when $$\frac{dr}{d\theta}$$<$$0.$$So solve:$$-3\sin\theta$$<$$0.$$$$\sin\theta$$>$$0.$$This happens on: $$(0,\pi).$$Final answer: $$(0,\pi)$$
On a given day, the number of people, in thousands, that have entered a stadium is modeled by $$h(t)=5.12\tan^{-1}(0.6t-0.3)$$, where $$t$$ is in hours and $$0 \le t \le 10$$. Based on the model, at what time $$t$$ did person number $$3000$$ enter the stadium?⌄
Person number $$3000$$ means $$h(t)=3$$Set the model equal to $$3$$.$$5.12\tan^{-1}(0.6t-0.3)=3$$Divide by $$5.12$$.$$\tan^{-1}(0.6t-0.3)=\frac{3}{5.12}$$Apply tangent to both sides.$$0.6t-0.3=\tan\left(\frac{3}{5.12}\right)$$$$0.6t-0.3\approx0.663688$$$$0.6t\approx0.963688$$$$t\approx1.606146$$Final answer: $$1.606$$
The function $$h$$ is given by $$h(x)=4\cdot3^x$$. For which value of $$x$$ is $$h(x)=972$$?⌄
Set the function equal to $$972$$.$$4\cdot3^x=972$$Divide by $$4$$.$$3^x=243$$Rewrite $$243$$ as a power of $$3$$.$$243=3^5$$So$$x=5$$Final answer: $$5$$
The function $$f$$ is given by $$f(x)=e^{3x}$$, and the function $$g$$ is given by $$g(x)=\ln(2x)$$ for $$x$$>$$0$$. Which of the following is an expression for $$f(g(x))$$?⌄
Substitute $$g(x)$$ into $$f$$.$$f(g(x))=e^{3\ln(2x)}$$Use the exponent rule.$$e^{3\ln(2x)}=e^{\ln((2x)^3)}$$$$e^{\ln((2x)^3)}=(2x)^3$$$$f(g(x))=8x^3$$Final answer: $$\text{B}$$
Consider the constant function $$f(x)=2$$ and the function $$g(x)=\log_2 x$$. Let $$h(x)=g(x)-f(x)$$. In the $$xy$$-plane, what is the $$x$$-intercept of the graph of $$h$$?⌄
The $$x$$-intercept occurs when $$h(x)=0$$Substitute.$$\log_2 x-2=0$$Solve.$$\log_2 x=2$$Rewrite in exponential form.$$x=2^2$$ $$x=4$$So, the point is $$(4,0)$$Final answer: $$(4,0)$$