Find the zeros of f (x)

(1)

(1) Find the limit if it exists.

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(a) lim

x→0⁺

√x p4 +√

x − 2 (b) lim

x→2

x|x − 2|

x − 2

(2) Take a sheet of rectangular paper with 5 cm long and 2 cm wide. Fold the lower left part of 8%

it so that the corner point touch the upper edge as shown in the figure. Find x such that the area A is maximized.

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A

B ...................................._{2 cm} ...

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(3) Let f (x) = 2x³+ 3x²− 2x. Find the zeros of f (x). Find the critical points and the inflection 18%

points of f (x). Where is f increasing, decreasing, concave up and concave down? Sketch the graph of f .

(4) Let g(x) = x f (x), where f is defined by 16%

f (x) =

(1, if x is rational;

0, if x is irrational.

(a) Determine whether g(x) is continuous at x = 0. Justify your answer.

(b) Determine whether g(x) is differentiable at x = 0. Justify your answer.

(5) Find an equation for the tangent line to the curve x²+ 4xy + y³+ 4 = 0 at the point (1, −1).

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(6) Determine whether the following is True or False. Justify your answer.

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(a) If limx→af (x) exists, then f (x) is continuous at x = a.

(b) If f (x) is continuous, then |f (x)| is continuous.

(c) If |f (x)| is continuous, then f (x) is continuous.

(d) If f (x) is continuous, then f (x) is differentiable.

(e) If f (x) is continuous on (a, b), then the absolute maximum value of f exists.

(f) Let f be defined by f (x) = x sin(_x¹) if x 6= 0, and f (0) = 0. Then f (x) is continuous at x = 0.

(g) Let f be defined by f (x) = x sin(¹_x) if x 6= 0, and f (0) = 0. Then f (x) is differentiable at x = 0.

(h) Let f (x) be continuous on an interval [a, b] and differentiable on (a, b). Then f is differ- entiable on [a, b].