1) Find the equation of the tangent line to f(x) = In(x) at x = e3. Use the tangent line to approximate f(20).
2) The cost of producing q pencils million pencils is given by C(q) = x/10 - ln(x)+ 10, where 10 C(q) is in cents. What is the marginal cost at q = 5? Interpret you answer. Should the pencil factory increase or decrease production?
3) The quantity q of cabbages sold by a fanner at a farmer's market depends on price p in dollars. The variables q and p are related by the equation q2 + 2p2 + 5xy = 2000. Find the elasticity of demand when p = 3. Interpret your answer. Is demand elastic, inelastic, or unitary at this price? Should the farmer increase or decrease the price of the cabbages?
4) Find the domain, all asymptotes, holes, extrema, and inflection points of the function f(x) = (2x - 2)/(3x2 + 3x - 6), as well as the intervals on which f(x) is increasing, decreasing, concave up, and concave down.
5) A large cone shaped funnel of height 5m and radius 3m contains a chemical mixture. The chemical mixture is being emptied out of the bottom of the funnel at a rate of 0.5 m3/sec. How fast is the depth of the chemical mixture changing when the depth is 3.5m?
6) Suppose that the state of Oregon can support at most 15 million people. In 2000 the population of Oregon was about 3.4 million. Today it is about 4 million. Create a logistic model for the population of Oregon. According to the model, what will the population be in 2020? When will the population reach 5 million? When will the population be growing the fastest?
7) Suppose you buy a car for $34,000. Its value depreciates at a continuous rate of 15% each year. How much will your car be worth after 5 years? When will your car be worth less than $1000? The first day you have the car, how fast is it kissing value (not the percent rate, just the rate of change)?
8) Use the definition of the derivative to find the derivative of f(x) = x2 - 3x + 1.
9) You by a plot of land whose value in t years is given by V(t) = 200000e n 3√t. If the prevailing interest rate is 7.7% and remains constant, when should you sell the land to maximize its present value?
10) Rigorously compute the limit as t goes to infinity of In (1 + 1/t . 5/1+e-t ) and justify each step using limit properties.