REVIEW EXERCISES FOR QUIZ 6 (sections 3.2, 3.3 and 3.4)
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Section 3.2: Monotonicity and concavity
Exercise 1: Determine where each of the following functions is increasing, decreasing, concave
up and concave down and find all the inflection points.
(i) f (x) = 2x2 + cos2 x
(ii) x2/3 (1 − x)
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Section 3.3: Local extrema
Exercise 2: For each of the following functions, find all the local extrema.
cos x
1+sin x ,
0 < x < 2π
(ii) f (x) = | sin x|,
0 < x < 2π
(i) f (x) =
Exercise 3: For each of the following functions, find all the local extrema. Determine also
the absolute extreme points.
(i) f (x) = sin x2 ,
(ii) f (x) = |x2 − 1|,
(iii) f (x) = x2 −
3
0≤x≤π
−2 ≤ x ≤ 2
16x2
,8
(8−x)2
<x<∞
Section 3.4: Optimization problems
Exercise 4: Find the equation of the line that is tangent to the ellipse
b2 x2 + a2 y 2 = a2 b2
in the first quadrant and forms with the coordinate axes the triangle with smallest possible area
(a and b are positive constants).
Exercise 5: A weight connected to a spring moves along the x-axis so that its x-coordinate
at time t is given by
√
x(t) = sin 2t + 3 cos 2t
What is the farthest that the weight gets from the origin?
Exercise 6: A fence h feet high runs parallel to a tall building and w feet from it. Find the
length of the shortest ladder that will reach from the ground across the top of the fence to the
wall of the building (see figure 1).
Exercise 7: See figure 2.
(i) A rectangle has two corners on the x-axis and the other two on the parabola y = 12 − x2 ,
with y ≥ 0. What are the dimensions of the rectangle of this type with maximum area?
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Figure 1: Exercise 6
Figure 2: Exercise 7
(ii) A rectangle is to be inscribed in a semicircle of radius r. What are the dimensions of the
rectangle if its area is to be maximized?
Exercise 8: One corner of a long narrow strip of paper is folded over so that it just touches
the opposite side, as shown in figure 3. With parts labeled as indicated, determine x in order
to:
(i) Maximize the area of triangle A.
(ii) Minimize the area od triangle B.
(iii) Minimize the length z.
Figure 3: Exercise 8
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