You will be graded on whether you complete this homework, not on how many questions you answer correctly. Questions of these types will appear in lectures and exams.

Evaluate and/or simplify the expressions without a calculator. If your answer is a fraction, leave it as a fraction. Complete this page before turning to the next page, which contains the answers.

QUESTION SET A

1A) (a²bc^-1)(ab^-3) = ?

2A) (a^2/3)^4/3 = ?

3A) log₁₀100 = ?

4A) log1 = ?

5A) simplify e^2ln3

6A) express as one term: ln 9 — ln 4 = ?

7A) use the quadratic equation to solve 2x² — x — 3 = 0

8A) For values of x which are close to 1, ln(x) ‰ x - 1. This problem uses a Taylor expansion

to show that this is true. Use a Taylor Series to approximate ln(1.05) = ? to first order.

9A) f(x) = (x^4/3 — 2x^-2)^{5 df}/_dx = ? (no need to multiply out)

10A) f(x) = (1 — x⁴)/x ^df/_dx = ? (no need to multiply out)

11A)

12A) find the maximum and minimum of the equation f(x) = x³ — 27x

13A)

14A)

15A)

Grade questions 1A-15A and make sure that you understand the answers before turning to the next page.

ANSWER SET A REASONS

1A) a³b^-2c^-1 a^xa^y = a^x+y

2A) a^8/9 (a^x)^y = a^xy

3A) 2 log_a(a^b) = b

4A) 0 log1 = log₁₀10⁰ = 0

5A) (e^ln3)² = (3)² = 9 e^lnx = x

6A) ln(9/4) = ln(3/2)² = 2ln(3/2) ln a — ln b = ln a + ln(b)^-1 = ln(a·b^-1) = ln(a/b),

7A) x = -1, ³/₂

8A) ln(x) = ln(a) + (x-a)(1/a) Taylor Series:

ln(1.05) = 0 + (1.05 — 1)(1/1) = 0.05 f(x) = f(a) + (x-a)f’(a) +(1/2!)(x-a)²f’’(a)…

9A) 5(x^4/3 — 2x^-2)⁴(⁴/₃x^1/3 + 4x^-3) for f(x) = cxⁿ, ^df/_dx = ncx^n-1, where c is a constant

10A) -4x³(¹/_x) + (1 — x⁴)(-1/x²) ^d/_dx(g(x)·h(x)) = (^d/_dxg(x))(h(x)) + (g(x))(^d/_dxh(x))

or use the quotient rule instead

11A) ¹/₂·¹/₃·¹/₄

12A) x = 3 is min, x = -3 is max min occurs at ^df/_dx = 0 and ^d2f/_dx2 > 0

max occurs at ^df/_dx = 0 and ^d2f/_dx2 < 0

13A)

14A)

Answer questions 1B-15B below without looking back to the "reasons" page.

QUESTION SET B

1B) 2³·4 = 2^?

2B) (a^3/5)³ = ?

3B) ln e³ = ?

4B) log₃81 = ?

5B) simplify 2e^3ln2

6B) express as one term: ln 3 + ln 4 = ?

7B) use the quadratic equation to solve x² + x + 2 = 0. x = ?

8B) For values of x which are close to 1, ln(x) ‰ x - 1. This problem uses a Taylor expansion

to show that this is true. Use a Taylor Series to approximate ln(17/16) = ? to first order.

9B) f(x) = (5x³ + x^2/3)^{4 df}/_dx = ? (no need to multiply out)

10B) f(x) = ^{(1 + x)}/_{(x2 — 2)} ^df/_dx = ? (no need to multiply out)

11B)

12B) Find the maximum and minimum of the equation f(x) = x³ — 48x

13B)

14B)

15B) �/�t (8st ^-2 — 3s^-3t⁴)

If you need more practice with differentiation and integrals, check out the worked problems at:

http://www.math.ucdavis.edu/~kouba/ProblemsList.html

1B) 2⁵

2B) a^9/5

3B) 3

4B) log₃(3⁴) = 4log₃3 = 4

5B) 2(e^ln2)³ = 2·2³ = 2⁴ = 16

6B) ln 12

7B)

8B) ln(¹⁷/₁₆) = ln(1) + (¹⁷/₁₆ — 1)(1/1) = ¹⁷/₁₆ — 1 = ¹/₁₆

9B) 4(5x³ + x^2/3)³ (15x² + ²/₃x^-1/3)

10B)

11B) 1·4·9 = 36

12B) x = 4 is min, x = -4 is max

13B)

14B)

15B) -16st^-3 — 12s^-3t³

Keep the first pages and turn in this page only. Name:

Did you get all questions 1A-14A correct on the first try? (circle an answer) YES / NO

Mark an "X" through the questions you answered incorrectly.

1A 2A 3A 4A 5A 6A 7A 8A 9A 10A 11A 12A 13A 14A 15A

1B 2B 3B 4B 5B 6B 7B 8B 9B 10B 11B 12B 13B 14B 15B