(1)Calculus I Practice problems Chap 1: Sec

Calculus I Practice problems 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. Compute lim_x→0 ^{1−cos 4x}_9x2 Ans: 8/9

3. Let f (x) =

( cx − 2 for x ≤ 2;

cx²+ 2 for x > 2 Find c such that f (x) is continuous.

Ans: c = −2

4. Determine the intervals on which f (x) = ln (1 − x²) is continuous. Ans: 1 − x²> 0 or (−1, 1) 5. Compute

(i) lim_x→0 ^√x+9−3_x (ii) lim_x→1−

2x x²−1

(iii) lim_{x→∞4x+999}^{x+sin x}

Ans: (i) 6 (ii) −∞ (iii) 1/4 Chap 2: Sec. 2.3-Sec. 2.9:

1. _dx^d [^x_3x+1^2−x] =?

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

3. (a) Let y = ln^q3x+1_5x+2. Find ^dy_dx. Ans: (3x−1)(x+1) (3x+1)²

(b) Let y = e^x2sin (x²+ x + 1) ·√

3x + 1/(x²− 1). Find ^dydx. Hint: Take ln on bouth side.

4. The equation 7x²y³− 5xy²− 4y = 7 defines y implicitly as a function of x. Find _dx^dy. Ans: _4+10xy−21x^14xy3−5y22y²

5. Find the detivative of f (x) = x^2x Ans: 2(ln x + 1)x^2x 6. Compute _dx^d cos⁻¹(2x³) Ans: √^−6x2

1−(2x³)²

7. 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 tan ((π/4) + 0.05) by the method of linear approximation (i.e., by differentials). Ans: 1+2∗(0.05) = 1.1

2. Compute lim_x→1^{+ ln x}

(x−1)² Ans: ∞ 3. Find the asymptotes of

(i) f (x) = ^(3x−1)_9x2 ²

−4 . Ans: V:x = 2/3, x = −2/3; H:y = 1 (ii) f (x) = ^(3x−1)_9x2 ²

−1 .Ans: V:x = −1/3; H:y = 1

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

4. 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

5. 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

6. Determine the concavity of f (x) = 4x³− x⁴. Ans: Concave up: (−∞, 0) ∪ (2, ∞); Concave down: (0, 2) 7. 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.

8. 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

9. 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 detivative of the revenue function, R(p) = p · D(p)

10. Compute:

(i) lim_x→0(_ln(x+1¹ −¹_x) (ii) lim_x→0⁺(cos x)^1/x (iii) lim_x→∞(1 +¹_x)^x

Ans (i) Sec 3.2 example 2.7 (ii) Sec 3.2 exercise 38 (iii) e Chap 4: Sec. 4.2-Sec. 4.7, Sec. 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 R_n.

Hint: 1 + 2 + ... + n = ⁿ⁽ⁿ⁺¹⁾₂

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

n→∞Rn (without using antiderivatives).

2. Find the derivatives of the following functions. It is not necessary to simplify your answer:

(a) f (x) = ((x²+ 1)³+ 1)⁴ (b) G(x) =^R₀^x2√

1 + t⁴dt

Ans: (a) 4((x²+ 1)³+ 1)³· 3 · (x²+ 1)²· (2x) (b)2x ·√ 1 + x⁸ 3. Let f be continuous and define F by

F (x) = Z x

0 [t² Z t

1 f (u) du] dt.

Find F⁰(x) and F⁰⁰(x). Ans: F⁰(x) = x^2R₁^xf (u) du, F⁰⁰(x) = 2x^R₁^xf (u) du + x²f (x) 4. Evaluate the given integral

(i) ^Rx(x + 1)⁹dx, Hint:u = x + 1 (ii)^R _sin2 ^{cos θ}

θ−2 sin θ−8dθ. Hint u = cos θ (iii) ^{R dx}

e^x√

4+e^2x. Hint: u = e^x, u = 2 tan θ (iv)^{R ln x}

x√

1+ln xdx, Hint:u = ln x (v) ^R √^x3

x²+1dx. Hint: x = tan θ (vi)^R_−∞^∞ _(1+e^e−₋^xx

)² dx Ans: 1

5. Evaluate the given integral (i) ^Rsec³t dt

(ii)^R sec t dt

Ans:(i) (− ln (cos t/2 − sin t/2) + ln (cos t/2 + sin t/2) + sec t tan t)/2 + C (ii) ln | sec t + tan t| + C 6. (a) ^R_−∞^∞ x³dx (b) lim_R→∞^R_−R^R x³dx Ans: (a) DIV (b) 0

7. Determine whether the integral converges or diverges:

(i) ^R₀¹x^−1/3dx (ii)^R₀¹x^−4/3dx (iii) ^R₁^∞x^−1/3dx

(iv)^R₋₁¹ x^−1/3dx Ans: (i) 3/2 (ii) DIV (iii) DIV (iv) 0 Chap 5: Sec. 5.1-Sec. 5.6.

1. Find the region bounded by the parabola x = 2 − y² and the line y = x.

Ans: ^R₋₂¹ (2 − y²) − y dy = ...

2. A solid is formed by revolving the circular disk (x − 5)²+ y² = 4 about the y-axis. Set up, but do not evaluate, a definite integral which give the volume of the solid.

Ans: ^R₋₂² π[(5 +^p4 − y²)²− (5 −^p4 − y²)²] dy or^R₃⁷2πx[^p4 − (x − 5)²− (−^p4 − (x − 5)²)] dx

3. Let Ω be the region bounded by y = sec x, x = 0, x = ^π₄ and y = 0. Find integrals represent the volume of the solids generated by Ω about (a) x-axis, (b) y-axis, (c) y = −1, (d) x = −1. (Don’t evaluate the integrals) Ans: Midterm 2

4. Set up a definite integral for the arc length of an ellipse x²+ 4y² = 4. Ans: 4^R₀^2r1 + (−₄√ ^x

1−x²/4)²dx or 4^R₀^π/2p(−2 sin θ)²+ (cos θ)²dθ

5. Set up the integral for the surface area of the surface of revolution. y = e^x, 0 ≤ x ≤ 1, revolved about x-axis.

Ans: S =^R₀¹2πe^xp1 + (e^x)²dx

6. (i) At time t, a particle has position x(t) = 1 − cos t, y(t) = t − sin t Find the total distance traveled from t = 0 to t = 2π. Find the speed of the particle at t = π.

Ans: ^R₀^2πp(sin t)²+ (1 − cos t)²dt, speed at t = π is 2

(ii) Find the area of the surface generated by revolving the curve y = cosh x, x ∈ [0, ln 2] about the x-axis.

Ans: S =^R₀^{ln 2}2π cosh x^p1 + (sinh x)²dx Chap 6: Sec. 6.1-Sec. 6.5.

1. Two years ago, there were 4 grams of a radioactive substance . Now there are 3 grams. How much was there 10 years ago? Ans: ⁴₃⁵4

2. Find the size of permanent endowment needed to generate an annual $2,000 forever at 10% (annual) interest compounded continuously. Ans: 2000 · e^−0.1

3. Solve the IVP, explicitly, if possible y⁰ = ^x−1_y2 , y(0) = 2. Ans: y = (³₂x²− 3x + 8)^1/3