MATH 141 Fall 2016 Review Exam

Not bad selection of problems.

The first group missed a couple of questions about solving exponential or logarithmic equations, and maybe some basic problems about fundamental properties of trigonometric functions. But your choice of problems was certainly interesting, and measure well the other relevant skills for this part of the course. A solid 4/5

The second group completely missed the point, except for a couple of questions on derivatives of basic functions. In my final exam you have to expect many more different derivatives. In particular, you should spend some time working on

  • The chain rule (explore exponential and logarithmic settings)
  • More implicit differentiation examples.
  • At least two different cases of logarithmic differentiation.
  • Derivatives of inverse functions
All in all, the second group will receive a 1/5.

The selection of questions for the third and fourth groups was outstanding. All the relevant topics are present, and all different skills are properly addressed. I wish I had seen at least the sketching of a rational function, but your choice is decent. Both groups receive a 5/5.

  1. Find the domain of \( f \circ g \) and \( g \circ f \) for the functions
    \begin{equation} f(x) = \sqrt{x+5}, \qquad g(x) = 1/x \end{equation}
  2. Given \( f(x) = 3 + \sin x, \) find the expression of a function obtained from \( f \) by performing a vertical stretch with factor 4.

  3. Find domain and range of the function \( f(x) = \dfrac{x-1}{x+2}. \)

  4. Study the continuity of the function \( f(x) = \dfrac{4x+10}{x^2-2x-15}. \)

  5. Compute the derivatives of the following functions:
    \begin{align} &f(x) = 2^x \cos x + \frac{5\ln x}{4x^2} && f(x) = \sin (x^2) \\ &f(x) = (x^3 -2x^2 +1)^3 && y = (x^2+4)^{e^{3x}} \end{align}
  6. Find \( \frac{dy}{dx} \) for a function \( y = y(x) \) that satisfies the implicit equation \( e^{x+2y} = 1. \)

  7. Compute the following limits:
    \begin{align} &\lim_{x \to 1} \frac{4x - \sin x}{1-x^3} && \lim_{x \to 2} - \frac{x-2}{x^2-4} \\ &\lim_{x \to 2^-} \frac{x^2 - 4x +3}{x^2 - 3x +2 } && \lim_{x \to \infty} \frac{3x^3+2x^2+4x}{8x^3+x^2+x} \\ &\lim_{x \to 3} \frac{\sqrt{x+1}-2}{x-3} && \lim_{x \to \infty} \frac{\ln (x^3+6x^2+3x+9)}{\ln x} \end{align}
  8. Find an equation for the tangent line to the function \( f(x) = \sqrt{x^2+3} \) at the point \( (-1,2). \)

  9. Find an equation for the normal line to the function \( f(x) = x^2 + \sin x \) at \( x = 0. \)

  10. A conical cup with a radius of 2.25 inches is filled with 6 inches of water. The cup begins to leak and the water level decreases at a rate of 1.5 inches per minute. After two minutes, at what rate is the volume of water changing?

  11. A box with an open top is to be constructed from a square piece of cardboard 5 ft wide by cutting out small squares from each of the 4 corners and bending up the sides. Find the largest volume that such a box can have and its surface area.

  12. A rectangular poster is being made, containing 64 square inches of printing and a 4-inch margin at the top and bottom, as well as a 1-inch margin along each side. What overall dimensions will minimize the amount of paper used to create this poster?

  13. At 3:00 PM dragon A is 100 miles east of dragon B. Dragon A is flying south at a rate of 10 miles per hour. Dragon B is flying north at a rate of 30 miles per hour. How fast is the distance between the two dragons changing at 7:00 PM?

  14. Find the dimensions of a rectangle with an area of 34300 square meters for which the perimeter is as small as possible.

  15. Sketch the graph of the function \( f(x) = e^x/x. \)

  16. Find and classify all critical points of the function \( f(x) = x^4 - 4x^3 + 10. \)

  17. Find all inflection points of the function \( f(x) = 2x^5-x^3+5x^2+x-12. \)

  18. Compute the following antiderivatives:
    \begin{align} &\int x-3 + \frac{6}{3x+2}\, dx && \int 3\cos x -4 \sin x +\sec^2 x\, dx && \int \frac{31x^2 -12x +5}{x}\, dx \\ & \int \frac{\ln(x)^{12}}{x}\, dx && \int \frac{dx}{25+x^2} && \int_{50}^{100} 5x e^{10x^2}\, dx \\ \end{align}
  19. Find the area and signed area of the function \( f(x) = x^2-4 \) over the interval \( [-3,3]. \)

  20. Compute the following partial sum:
    \begin{equation} \sum_{n=1}^{1000} (5-4n+7n^2) \end{equation}
  21. Compute \( \displaystyle{\lim_{N \to \infty} \frac{2}{N^3} \sum_{n=1}^N (n^2+4n+9)}. \)

  22. Find the general term of the sequence with first three terms being \( -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \dotsc \)