M119 Review Questions

  1. Find $f'(x)$ given $$f(x)=2e^{\sqrt x}+\ln x^3$$.
  2. Evaluate $$\lim_{h\to 0}{\ln(2+h)-\ln 2\over h}$$.
  3. Find the area bounded by the curves $y=x^2$, $y=5$, and $y=25x$.
  4. Determine all relative extrema of $$y={x\over x^2+9}$$.
  5. A poster is to have an area of 125 square inches. The printed material is to be surrounded by a margin of 3 inches at the top and margins of 2 inches at the bottom and sides. Find the dimensions of the poster that maximizes the area of the printed material.
  6. Suppose $f(3)=5$ and $f'(x)=e^{5x}$. What is $f(2)$?
  7. Find $$\int {\sqrt{\ln x}\over x} dx$$.
  8. Find $$\lim_{x\to\infty}{3x^2+12x-9\over 5x^2+8}$$.
  9. Suppose $ye^x+xe^y=1$. What is the slope of this curve at the point $(x,y)=(1,0)$?
  10. You deposited 100 dollars in a bank account some time ago and now it is worth 350 dollars. If the interest rate was 8% compounded continuously, then how many years ago did you deposit the 100 dollars?
  11. Use a Riemann sum with $n=5$ to approximate the area under the graph of the hyperbola $y=1/x$ from $x=1$ to $x=2$. Use the left endpoints in constructing the rectangles.
  12. Where is the graph of $$y=e^{x^3}$$ increasing, decreasing, concave up, and concave down?
  13. Evaluate $$\int_0^5 (x^2+5x+17)\,dx$$.
  14. If $f(x)=\ln\big[(x^2+5)^2(x^5+7)^5\big]$, what is $f'(1)$?