Calculus I, 2009

Calculus I, 2009 Practice problems

• Limits and Continuity:

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) . 2. Compute lim

x→0

1− cos 4x 9x² 3. Let f (x) =

{ cx− 2 for x ≤ 2;

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

4. Determine the intervals on which f (x) = ln (1− x²) is continuous.

5. Compute (i) lim

x→0

√x + 9− 3 x (ii) lim

x→1⁻

2x x²− 1 (iii) lim

x→∞

x + sin x 4x + 999

• Differentiation:

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

2. Let y = e^x2sin (x²+ x + 1)·√

3x + 1/(x²− 1). Find dy dx.

3. The equation 7x²y³− 5xy²− 4y = 7 defines y implicitly as a function of x. Find ^dy_dx.

4. Find the detivative 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)

5. Compute _dx^d cos⁻¹(2x³)

6. Determine if f (x) = x⁷+ 2x³−2006 is increasing, decreasing or neither. Prove f(x) = 0 has exactly one solution.

• Application of Differentiation:

1. Estimate tan ((π/4) + 0.05) by the method of linear approximation (i.e., by differen- tials).

2. Compute lim

x→1⁺

ln x (x− 1)² 3. Find the asymptotes of

(i) f (x) = (3x− 1)²

9x²− 4 . (ii) f (x) = (3x− 1)²

9x²− 1 .(iii)f (x) = (3x− 1)² x− 1 4. Let f (x) = 2x³− 3x²− 12x. Find the relative extrema of f(x).

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5. Find the absolute maximum and minimum values of the function f (x) = 2x³−9x²+12x over the interval [0, 2].

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

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.

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?

10. Compute:

(i) lim

x→0( 1

ln(x + 1− 1 x) (ii) lim

x→0⁺(cos x)^1/x (iii) lim

x→∞(1 + 1 x)^x

• Integration:

1. Let f (x) = x + 1

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

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

n→∞R_n(without using antideriva- tives).

2. Find the derivatives of the following functions. It is not necessary to simplify your answer: G(x) =

∫ _x²

0

√1 + t⁴dt

3. Let f be continuous and define F by

F (x) =

∫ _x

0

[t²

∫ _t

1

f (u) du] dt.

Find F⁰(x) and F⁰⁰(x).

4. Evaluate the given integral (i)

∫

x(x + 1)⁹dx, (ii)

∫ cos θ

sin²θ− 2 sin θ − 8dθ. (iii)

∫ dx

e^x√

4 + e^2x. (iv)

∫ ln x x√

1 + ln xdx, (v)

∫ x³

√x²+ 1dx. (vi)

∫ _∞

−∞

e^−x (1 + e^−x)² dx 2

5. Evaluate the given integral (i)

∫ √

xe^√xdx; (ii)

∫ √ ln x

x dx; (iii)

∫ x

x + 4dx; (iv)

∫

(ln x)²dx;

6. Evaluate the given integral (i)

∫ ln x

x dx; (ii)

∫

ln (x²) dx; (iii)

∫ 3x

x²− 3x − 4dx; (iv)

∫ −2x²+ 4 x³+ 2x²+ xdx 7. Evaluate the definite integrals:

(i)^∫₁⁴√

xe^√xdx; (ii) ^∫₁^e(ln x)²dx;

8. Evaluate the given integral (i)

∫

sec³t dt (ii)

∫

sec t dt

9. (a)

∫ _∞

−∞x³dx (b) lim

R→∞

∫ _R

−Rx³dx

10. Determine whether the integral converges or diverges:

(i)

∫ ₁

0

x^−1/3dx (ii)

∫ ₁

0

x^−4/3dx (iii)

∫ _∞

1

x^−1/3dx (iv)

∫ ₁

−1x^−1/3dx

• Application of Integration

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

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.

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)

4. Set up a definite integral for the arc length of an ellipse x²+ 4y² = 4.

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

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.

(ii) Find the area of the surface generated by revolving the curve y = cosh x = e^x+ e^−x

2 ,

x∈ [0, ln 2] about the x-axis.

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