4: Exponential and Logarithmic Functions (80 marks total) Your mark will be given as a percentage. [10 marks] 1. 100 1 Consider the function . 1. Give exact (5 marks) a. Find an equation of the tangent line at values. (5 marks) b. If represents the percentage of people who have heard a rumour in months, when will 90% of people have heard the rumour? [10 marks] 2. ln 3 . Consider the function (5 marks) a. Find an equation of the tangent line at (5 marks) b. Find and simplify [16 marks] 3. a. (4 marks) b. (4 marks) c. (4 marks) d. 4. . Differentiate the following functions. You need not simplify your answer. (4 marks) [8 marks] 1. ln 5 5. 2 . ln 5 . ln . Suppose that a company finds that the percentage of people who bought their product after advertisements had been run on television is modelled by . . (2 marks) a. What percentage of people bought the product before seeing the advertisements? (3 marks) b. Find and interpret (3 marks) c. Find 5 and 10 . What do your answers tell you about the function ? . [10 marks] 5. Find the maximum and minimum values, giving exact values, of (5 marks) a. (5 marks) b. [10 marks] 6. over 0, 5 . ln over 1, 4 . Suppose that the total cost, in millions of dollars, for a company where is the time in years is given by 500 70 since its startup date. (3 marks) a. Find and interpret (3 marks) b. On which interval(s) is C is increasing and on which C is decreasing. (3 marks) c. On which interval(s) is C concave up and on which interval(s) it is concave down? (1 mark) d. Sketch the graph of C. [8 marks] 7. . Suppose that the population in a certain country was 150 million in 1990 and 140 million in 1995. Assuming an exponential decay model, (3 marks) a. Find the value of k and write the model, using t as the number of years since 1990. (2 marks) b. Estimate the population in 1998. (3 marks) c. In what year will the population be 136 million? [8 marks] 8. (2 marks) a. How much money should you put in an investment today to yield a lump sum of $100,000 in 30 years if interest is compounded continuously at an annual percentage rate of 2.75%? (3 marks) b. Suppose that the supply and demand for a given item are , given by and , where is the number of items that a company is willing to sell for dollars each and is the quantity that the public is willing to buy. Find the equilibrium point (that is the point at which supply equals demand). (3 marks) c. Make a sketch of both functions from part b. in the same graph and indicate the equilibrium point. T