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 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.