(1)Chap 1: Sec

Chap 1: Sec. 1.2-Sec. 1.5:

1. Let f (x) =







|x + 2| for x ≤ 0;

2 + x² for 0 < x < 2;

x³ for x≥ 2

. Find (a) lim_x→0−f (x), (b)lim_x→0⁺f (x), (c) lim_x→2−f (x), (d)lim_x→2⁺f (x), (e) lim_x→0f (x), (f) lim_x→2f (x) .

Ans: (a) 2 (b) 2 (c) 6 (d) 8 (e) 2 (f) DNE

2. Let f (x) = {

cx− 2 for x≤ 2;

cx²+ 2 for x > 2 Find c such that f (x) is continuous. Ans: c =−2 3. Determine the intervals on which f (x) = ln (1− x²) is continuous.

Ans: 1− x²> 0 or (−1, 1) 4. Compute (i) limx→0

√x+9−3

x (ii) lim_x→1− 2x x²−1

Ans: (i) 6 (ii)−∞

Chap 2: Sec. 2.3-Sec. 2.9:

1. Find the tangent line to the curve y = x³− 4x²+ 2x + 1 at the point (1, 0).

Ans: ^dy_dx = 3x²− 8x + 2; slope=−3; y − 0 = −3(x − 1) 2. Let y = e^x2 · (x²+ x + 1)·√

3x + 1/(x²− 1). Find ^dy_dx. Hint: Take “ln” on both side.

3. The equation 7x²y³− 5xy²− 4y = 7 defines y implicitly as a function of x. Find _dx^dy. Ans:_4+10xy^14xy3_−21x^−5y22y²

4. Find the derivative of (i) f (x) = x^2x; (ii) g(x) = ^x_3x+1^2−x; (iii) h(x) = ln

√3x+1

5x+2 (assuming 3x + 1 > 0)

Ans: (i)2(ln x + 1)x^2x;(ii) ^(2x−1)(3x+1)−(x²−x)(3)

(3x+1)² = ^(3x_(3x+1)^−1)(x+1)2 ; (iii)= ¹₂(_3x+1³ − _5x+2⁵ )

5. Determine if f (x) = x⁷+ 2x³− 2006 is increasing, decreasing or neither. Prove f(x) = 0 has exactly one solution. Hint: Sec. 2.9 example 9.1

Chap 3: Sec. 3.1-Sec. 3.8:

1. Estimate √³

8.02 by the method of linear approximation (i.e., by differentials).

Ans: f (x) = x^1/3;f (x)≈ f(8) + f⁰(8)(x− 8);√³

8.02≈ 2 +₁₂¹(0.02)≈ 2.001667 2. Find the asymptotes of

(i) f (x) = ^(3x_9x⁻¹⁾2−4². Ans: V:x = 2/3, x =−2/3; H:y = 1 (ii) f (x) = ^(3x_9x⁻¹⁾2−1².Ans: V:x =−1/3; H:y = 1

(iii)f (x) = ^(3x_x⁻¹⁾₋₁² Ans: V:x = 1; H:none; S:y = 9x + 3.

3. Let f (x) = 2x³− 3x²− 12x. Find the relative extrema of f(x).

Ans: local max at x =−1; local min at x = 2; no abs. max/min

4. Find the absolute maximum and minimum values of the function f (x) = 2x³− 9x²+ 12x over the interval [0, 2]. Ans: Abs max: f (1) = 5; abs min: f (0) = 0

5. Determine the concavity of f (x) = 4x³− x⁴.

Ans: Concave up: (−∞, 0) ∪ (2, ∞); Concave down: (0, 2)

6. If 300 cm² of material is available to make a box with square base and an open top, find the largest possible volume of the box. Explain why your answer is the absolute maximum.

Hint:example 6.2

7. Sketch the graph of the continuous function f that satisfies the conditions:

f⁰⁰(x) > 0 if|x| > 2, f⁰⁰(x) < 0 if|x| < 2;

f⁰(0) = 0, f⁰(x) > 0, if x < 0, f⁰(x) < 0, if x > 0;

f (0) = 1, f (2) = 1

2, f (x) > 0 for all x, and f is and even function.

Ans: The graph looks like e^−x2/8

8. An automobile dealer is selling cars at a price of $12,000. The demand function is D(p) = 2(15− 0.001p)², where p is the price of a car. Should the dealer raise or lower the price to increase the revenue? Hint: Find the derivative of the revenue function, R(p) = p· D(p) 9. Compute:

(i) lim_x→0(_ln(x+1)¹ −¹_x); (ii) lim_x→1⁺ _(x^{ln x}₋₁₎2

Ans (i) Sec 3.2 example 2.7 (ii) Sec 3.2 exercise 38 Chap 4: Sec. 4.2-Sec. 4.8 (Integration Tables), 4.10,

1. Let f (x) = x + 1

(a) Divide the interval [0, 5] into n equal parts, and using right endpoints find an expression for the Riemann sum Rn.

Hint: 1 + 2 + ... + n = ⁿ⁽ⁿ⁺¹⁾₂

(b) Using the answer you got from part(a), calculate lim

n→∞R_n(without using antiderivatives).

2. Evaluate the given integral (i) ^∫x(x + 1)⁹dx, Hint:u = x + 1 (ii)^{∫ dx}

e^x√

4+e^2x. Hint: u = e^x,積分表 (iii) ^{∫ ln x}

x√

1+ln xdx, Hint:u = ln x (iv)^∫ √^x3

x²+1dx. Hint: u = 1 + x²

3. Evaluate the given integral (i) ^∫√

xe^√xdx =?; (ii)^{∫ √ln x}_x dx =?; (iii)^∫ _x+4^x dx =?; (iv)^∫(ln x)²dx =?;

Hint: (i) u =√

x;(ii) u = ln x; (iii) = x− 4 ln |x + 4|; (iv) u = (ln x)²;dv = dx 4. Evaluate the given integral

(i) ^{∫ ln x}_x dx =?; (ii)^∫ln (x²) dx =?; (iii)^∫ _x2−3x−4^3x dx =?; (iv) ^∫ _x^−2x3+2x²⁺⁴²+xdx =?

Hint: (i) u = ln x;(ii)u = ln x²;dv = dx; (iii) x²− 3x − 4 = (x − 4)(x + 1);

(iv) x³+ 2x²+ x = x(x + 1)² 5. Evaluate the definite integrals:

(i) ^∫₁⁴√

xe^√xdx =?; (ii)^∫₁^e(ln x)²dx =?;

6. Evaluate the given integral

i)^∫_−∞^∞ _(1+e^e−x_−x)2 dx ^∫_−∞^∞ x³dx iii) lim_R→∞^∫_−R^R x³dx Ans: (i) 1 (ii) DIV (iii) 0

7. Determine whether the integral converges or diverges:

i)^∫₀¹x^−1/3dx ii)^∫₀¹x^−4/3dx iii) ^∫₁^∞x^−1/3dx iv)^∫₋₁¹ x^−1/3dx Ans: (i) 3/2 (ii) DIV (iii) DIV (iv) 0