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. (i) limx→0√x+9x−3; (ii) limx→1− 2x
x2−1; (iii) limx→∞x+sin x4x+999
• Differentiation:
1. Find the tangent line to the curve y = x3− 4x2+ 2x + 1 at the point (1, 0).
2. Let y = ex2sin (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 dydx.
4. Find the detivative of (i) f (x) = x2x; (ii) g(x) =x3x+12−x; (iii) h(x) = ln
√3x+1
5x+2 (assuming 3x + 1 > 0) 5. Compute dxd 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 differentials).
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).
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:
f′′(x) > 0 if|x| > 2, f′′(x) < 0 if|x| < 2;
f′(0) = 0, f′(x) > 0, if x < 0, f′(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. (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
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.
(b) Using the answer you got from part(a), calculate lim
n→∞Rn (without using antiderivatives).
2. Find the derivatives of G(x) =
∫ x2
0
√1 + t4dt. It is not necessary to simplify your answer:
3. Let f be continuous and define F by
F (x) =
∫ x
0
[t2
∫ t
1
f (u) du] dt.
Find F′(x) and F′′(x).
4. Evaluate the given integral (i)
∫
x(x + 1)9dx, (ii)
∫ cos θ
sin2θ− 2 sin θ − 8dθ. (iii)
∫ dx
ex√
4 + e2x. (iv)
∫ ln x
x√
1 + ln xdx, (v)
∫ x3
√x2+ 1dx. (vi)
∫ ∞
−∞
e−x (1 + e−x)2dx 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)∫4 1
√xe√xdx; (ii)∫e
1(ln x)2dx;
8. Evaluate the given integral (i)
∫
sec3t dt; (ii)
∫
sec t dt;
9. (a)
∫ ∞
−∞
x3dx (b) lim
R→∞
∫ R
−R
x3dx
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
−1
x−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 = ex, 0 ≤ x ≤ 1, revolved about x-axis.
6. Find the area of the surface generated by revolving the curve y = cosh x = ex+ e−x
2 , x∈ [0, ln 2] about the x-axis.
7. Let f (x) = 643 x√
16− x2for 0≤ x ≤ 4 and f(x) = 0 for other values of x.
(a) Verify that f is a probability density function.
(b) Find P (X < 2).
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