Exam 3

On this exam you may use your calculator to check yourself, to find derivatives, and to solve equations. But you must show work. (1) Evaluate: $\sum_{i=2}^{5} \frac{2}{i}$

(2) Evaluate:

(a) $\int \frac{2x^2+x}{\sqrt{x}} dx$

(b) $\int f'(x) dx$

(3) Use Newton's Method to estimate $\sqrt{5}$ using the function f(x) = x²-5.

(4) Find a linear approximation to the curve $y= {\sin}^2x - {\cos}^2x$ at the point (0,-1).

(5) A farmer estimates that her farm would yield 1000 bushels of apples right now. Today's price per bushel is $10.00, but it will drop $0.05 per day from on. On the other hand, she guesses that the crop yield will be 10 more bushels per day. In how many days from now should she harvest her crop in order to maximize revenue?

(6) Consider the right triangle on the back cover of your text. Label the hypotenuse with x, the opposite side y, and the adjacent side z.

(a) What is the domain for $\theta$.

(b) Find the value of $\theta$ which maximizes the perimeter if x = 4. (Hint: Use the sine and cosine functions to write y and z in terms of x)

(7) Determine where the graph of f(x) = x⁴ - 6x³ + 12x² + 14x +3 is concave up. What are the inflection points of the graph?

(8) Problem 1 on Exam 2 - Graphing and Optimization with the Derivative