MATH 122 Fall 2014 Review Exam (24)
Assessment
 What is the effect of the operation \( f(x)+3 \) on the graph of the function \( f(x) \)?
 Write the expression of a function obtained by vertically stretching the graph of \( y=x^4 \) by a factor of 3, followed by a horizontal shift left of 2 units.
 Is the function \( f(x) = (5x)^3 \) a power function? If so, write it in the form \( f(x) = K x^n \).
 Estimate the instantaneous rate of change of the function \( f(x)=x\ln x \) between \( x=1 \) and \( x=2 \).
 Find the derivative of the following functions:
\begin{align} p(t) &= e^{0.03t} \\ f(x) &= 2(3x+5)^3 \\ f(x) &= (x7x^7)(\sqrt{x}+5) \\ f(x) &= \frac{12x^2}{4x^3+7} \\ f(x) &= 7^x + 2x^4 \\ C(t) &= \frac{e^{2t}}{t} \end{align}
 Compute the second derivative of \( h(x) = \ln (3x^24) \).

The following table gives the percentage of the US population in urban areas as a function of the year
Year 1800 1830 1860 1890 1920 1950 1980 1990 2000 Percentage 6.9 8.7 17.4 36.0 51.5 66.8 73.7 75.7 80.1  Find the average rate of change of the percentage of population living in urban areas from 1890 to 1990.
 Estimate the rate at which this percentage is increasing in 1990.
 Estimate the rate of change of this function for the year 1830, and explain what this means.
 Is this an increasing or decreasing function?
 Use a small interval (\( x=2 \) to \( x=2.01 \)) to estimate \( f’(2) \) for the function \( f(x) = x^6 e^{3x} \).
 Find an equation of the tangent line to the graph of \( f(x) = x^2e^{x} \) at \( x=0 \).
 The quantity demanded of a certain product, \( q \), is given in terms of the price \( p \), by the formula
\begin{equation} q = 1000e^{0.02p} \end{equation}
 Write the revenue \( R \), as a function of the price.
 Find the rate of change of the revenue with respect to the price.
 Find revenue and rate of change of revenue with respect to price, when the price is $10. Interpret this answer in financial terms.