124 C H A P T E R 4 / F U N C T I O N S O F O N E V A R I A B L E6. Which of the following formulas are always true and which are sometimes false (all variablesare positive)? (a) lnA + BC= ln A + ln B – ln C(b) lnA + BC= ln(A + B) – ln C (c) lnA B+ lnB A= 0 (d) p ln(ln A) = ln(ln Ap)(e) p ln(ln A) = ln(ln A)p (f) ln Aln B + ln C= ln A(BC)-17. Simplify the following expressions:(a) exp!ln(x)” – ln!exp(x)” (b) ln!x4 exp(-x)” (c) exp!ln(x2) – 2 ln y”R E V I E W P R O B L E M S F O R C H A P T E R 41. (a) Let f (x) = 3 – 27×3. Compute f (0), f (-1), f (1/3), and f (√3 2).(b) Show that f (x) + f (-x) = 6 for all x.2. (a) Let F (x) = 1 + 4xx2 + 4 . Compute F(0), F(-2), F(2), and F(3).(b) What happens to F (x) when x becomes large positive or negative?(c) Give a rough sketch of the graph of F.3. Figure A combines the graphs of a quadratic function f and a linear function g. Use the graphsto find those x where: (i) f (x) ≤ g(x) (ii) f (x) ≤ 0 (iii) g(x) ≥ 0. y––y =f (x)–2–321123 4xy = g(x) 4316 54 3 2 1—Figure A4. Find the domains of:(a) f (x) = %x2 – 1 (b) g(x) = √x1- 4 (c) h(x) = %(x – 3)(5 – x)5. (a) The cost of producing x units of a commodity is given by C(x) = 100 + 40x + 2×2.Find C(0), C(100), and C(101) – C(100).(b) Find C(x + 1) – C(x), and explain in words the meaning of the difference.R E V I E W P R O B L E M S F O R C H A P T E R 4 1256. Find the slopes of the straight lines (a) y = -4x + 8 (b) 3x + 4y = 12 (c) xa+y b= 17. Find equations for the following straight lines:(a) L1 passes through (-2, 3) and has a slope of -3.(b) L2 passes through (-3, 5) and (2, 7).(c) L3 passes through (a, b) and (2a, 3b) (suppose a 0).8. If f (x) = ax + b, f (2) = 3, and f (-1) = -3, then f (-3) = ?9. Fill in the following table, then make a rough sketch of the graph of y = x2ex. x-5-4-3-2-101y = x2ex 10. Find the equation for the parabola y = ax2+bx+c that passes through the three points (1, -3),(0, -6), and (3, 15). (Hint: Determine a, b, and c.)11. (a) If a firm sells Q tons of a product, the price P received per ton is P = 1000 – 1 3 Q. Theprice it has to pay per ton is P = 800 + 1 5 Q. In addition, it has transportation costs of 100per ton. Express the firm’s profit π as a function of Q, the number of tons sold, and findthe profit-maximizing quantity.(b) Suppose the government imposes a tax on the firm’s product of 10 per ton. Find the newexpression for the firm’s profits πˆ and the new profit-maximizing quantity.12. In Example 4.6.1, suppose a tax of t per unit produced is imposed. If t < 100, what productionlevel now maximizes profits?13. (a) A firm produces a commodity and receives $100 for each unit sold. The cost of producingand selling x units is 20x+0.25×2 dollars. Find the production level that maximizes profits.(b) A tax of $10 per unit is imposed. What is now the optimal production level?(c) Answer the question in (b) if the sales price per unit is p, the total cost of producing andselling x units is αx + βx2, and the tax per unit is t.⊂SM ⊃14. Write the following polynomials as products of linear factors: (a) p(x) = x3 + x2 – 12x(b) q(x) = 2×3 + 3×2 – 18x + 815. Which of the following divisions leave no remainder? (a and b are constants; n is a naturalnumber.)(a) (x3 – x – 1)/(x – 1)(c) (x3 – ax2 + bx – ab)/(x – a)(b) (2×3 – x – 1)/(x – 1)(d) (x2n – 1)/(x + 1) 16. Find the values of k that make the polynomial q(x) divide the polynomial p(x):(a) p(x) = x2 – kx + 4; q(x) = x – 2 (b) p(x) = k2x2 – kx – 6; q(x) = x + 2(c) p(x) = x3 – 4×2 + x + k; q(x) = x + 2 (d) p(x) = k2x4 – 3kx2 – 4; q(x) = x – 1126 C H A P T E R 4 / F U N C T I O N S O F O N E V A R I A B L E⊂SM ⊃17. The cubic function p(x) = 1 4×3 – x2 – 11 4 x + 15 2 has three real zeros. Verify that x = 2 is oneof them, and find the other two.18. In 1964 a five-year plan was introduced in Tanzania. One objective was to double the real percapita income over the next 15 years. What is the average annual rate of growth of real incomeper capita required to achieve this objective?⊂SM ⊃19. Figure B shows the graphs of two functions f and g. Check which of the constants a, b, c, p,q, and r are > 0, = 0, or < 0.yxyxy = f (x) =ax + bx + cy = g(x) = px2 + qx + rFigure B20. (a) Determine the relationship between the Celsius (C) and Fahrenheit (F) temperature scaleswhen you know that (i) the relation is linear; (ii) water freezes at 0◦C and 32◦F; and(iii) water boils at 100◦C and 212◦F.(b) Which temperature is represented by the same number in both scales?21. Solve for t: (a) x = eat+b (b) e-at = 1/2 (c) √12π e- 1 2 t2 = 1 8⊂SM ⊃22. Prove the following equalities (with appropriate restrictions on the variables):(a) ln x – 2 = ln(x/e2) (b) ln x – ln y + ln z = ln xzy(c) 3 + 2 ln x = ln(e3x2) (d) 12ln x –3 2ln1 x– ln(x + 1) = lnx2x + 1

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