Questions Sample for the Math 103 Final Exam May 2007
Calculators are not permitted on the actual nal exam. The actual nal exam will have fewer question; these problems are intended as a study guide.
1. Draw an accurate graph of the function f (x) = 2x + 5. Show the intercepts and write their coordinates on the graph.
6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 1 2 3 4 5 6
Find the slope-intercept form for the equation of the line whose graph is shown below:
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3
2
1
-4
-3
-2
-1 -1
1
2
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4
-2
-3
-4
1
2. Find the derivatives of the following functions. (a) f (x) = 2x3 3x + 7 (b) g(x) = ln(x) x (Simplify this derivative.) 2x2 x + 3 (d) k(x) = e2x+7 (c) h(x) = (e) r(x) = x ln(x) (f) s(x) = ex (g) t(x) = (2x4 x3 + 3x2 7)(x3 + x2 2)
3. Find the equation for the line tangent to the curve f (x) at the point (1, 9): f (x) = 3x2 x + 7
4. Find the absolute maximum and minimum values for the function f (x) = x3 6x2 + 9x + 1, on the interval [0, 2].
5. Suppose we have an $6000 to invest. (a) What amount will our account have after 10 years if is invested at an annual rate of 3% compounded semi-annually. (b) How long will it take for our account to grow to $12,000 if it is invested at an annual rate of 3% compounded continuously.
2
6. Consider the following function. x3 , x + 3, f (x) = 7, if x 2 if 2 < x < 4 if x 4
(a) Where is this function discontinuous? Why? (b) Where is this function dierentiable? Why?
7. A company manufactures and sells x units per week. The weekly price demand and cost functions are: x p(x) = 4 50 C(x) = 25 + x. (a) Find the production level that will maximize prot and the price the company should charge for each unit. (b) Explain in words why the value x for which prot is maximized is the same as the x value for which the slope of the tangent line to the revenue function is the same as that of the cost function. (c) If the xed production costs increase from 25 to 30, will your answer to (a) change? How? Why? (Note: You do not have to compute here.)
8. Continuing with a company manufactures and sells x units per week. The weekly price demand and cost functions are: p(x) = 4 x 50
C(x) = 25 + x. Graph the revenue and cost functions. You do not have to nd the break-even points. (But it is not a bad idea to do this for practice.) (a) Label the axes and mark the scale (tick marks) on the axes; 3
(b) Label y = R(x), its x-intercepts, its maximum point with coordinates; (c) Label y = C(x), its y-intercept and one other point on y = C(x). (d) Shade the areas which correspond to the company making a prot make a bold line to represent the maximum prot graphically.
100
75
9. Continuing with a company manufactures and sells x units per week with weekly pricex demand equation: p = 4 50 (a) Use the price-demand equation to solve for x in terms of p to get a function f (p) which gives the demand as a function of the price p. (b) Compute the elasticity of demand function for this price-demand function. (c) At p = $3: a price increase of 10% will create a demand decrease of what percent? (d) Is the revenue function or increasing decreasing at a price of $3 per unit?
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10. A company manufactures and sells x units per week. The price demand equation is of the form p = ax + b. When the price is $7 the demand is 2 units and when the price is $9, the demand is 1 unit. Find a and b. Suppose that the supply x and price p are related by the equation p = 2x + 3. (a) Find the equilibrium price and quantity. (b) Find the augmented matrix that represents this system of equations you solved in (a). Remark: Other questions here could be: Find the supply function instead of nd the demand function. Solve the system of equations using matrices. Graph the lines. Look also at problems 61-66 in 4.4 homework problems in text (I would use easier numbers).
11. Alpha Metal Works is considering producing and selling aardvark cages. The research department estimates that the xed costs to retool and the manufacture the new cages will be $1,200 and the variable costs will be $10 per cage. (a) Write an algebraic expression for the cost function to produce x cages: C(x) = (b) The price-demand equation for the aardvark cages is p + (0.6)x = 120. Price is given in dollars, and x is the demand at price p. Write an algebraic expression for revenue as a function of demand. R(x) =
12. Beta Borax Inc. plans to introduce a new shower cleaner. The cost (in dollars) to produce x tons of cleaner is C(x) = 3000 + 25x. The price-demand equation is p = 100 (0.5)x. 5
(a) Write an expression for revenue as a function of demand: R(x) = (b) Compute the marginal cost and marginal revenue functions: Marginal cost: Marginal revenue: (c) Use the marginal revenue function to approximate the revenue for selling the 11th ton of cleaner.
13. Gamma Gum Works makes and sells chewing gum. The price-demand equation is x = 100 2p, where x is the demand (in pounds) for chewing gum at a price of p (in ...