Assessment

1. What is the effect of the operation \( -f(x)+3 \) on the graph of the function \( f(x) \)?
2. 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.
3. Is the function \( f(x) = (5x)^3 \) a power function? If so, write it in the form \( f(x) = K x^n \).
4. Estimate the instantaneous rate of change of the function \( f(x)=x\ln x \) between \( x=1 \) and \( x=2 \).
5. Find the derivative of the following functions:
   \begin{align} p(t) &= e^{0.03t} \\ f(x) &= 2(3x+5)^3 \\ f(x) &= (x-7x^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}
6. Compute the second derivative of \( h(x) = \ln (3x^2-4) \).

7. 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?
8. Use a small interval (\( x=2 \) to \( x=2.01 \)) to estimate \( f’(2) \) for the function \( f(x) = x^6 e^{3x} \).
9. Find an equation of the tangent line to the graph of \( f(x) = x^2e^{-x} \) at \( x=0 \).
10. 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.