A Mathematical Orchard: Problems and Solutions 1st. Edition by Mark I. Krusemeyer, George T. Gilbert, Loren C. Larson | 9780883858332 | 0883858339 | Holooly

A Mathematical Orchard: Problems and Solutions

208 SOLVED PROBLEMS

Question: P.177

Let c =∑n=1^∞ 1/n(2^n − 1) = 1 +1/6+1/21+1/60+ · · ·. Show that e^c =2/1·4/3·8/7·16/15· · · · . ...

Verified Answer:

Let

P_k

be the partial product [lat...

Question: P.176

Let n ≥ 3 be a positive integer. Begin with a circle with n marks about it. Starting at a given point on the circle, move clockwise, skipping over the next two marks and placing a new mark; the circle now has n + 1 marks. Repeat the procedure beginning at the new mark. Must a mark eventually appear ...

Verified Answer:

To show why, suppose there is a pair

m_1, m...

Question: P.175

Let f be a continuous function on [0, 1], which is bounded below by 1, but is not identically 1. Let R be the region in the plane given by 0 ≤ x ≤ 1, 1 ≤ y ≤ f(x). Let R1 = {(x, y) ∈ R|y ≤ y} and R2 = {(x, y) ∈ R|y ≥ y}, where y is the y-coordinate of the centroid of R. Can the volume obtained by ...

Verified Answer:

Solution 1. To find an example, we let R be the tr...

Question: P.174

Let x0 be a rational number, and let (xn)n≥0 be the sequence defined recursively by xn+1 =|2xn³/3xn² − 4|. Prove that this sequence converges, and find its limit as a function of x0. ...

Verified Answer:

\lim_{n\to\infty}x_{n}={\left\{\begin{array...

Question: P.173

Find lim n→∞ ∫0^∞ n cos( √^4 x/n²)/1 + n²x²dx. Idea. First, we note that we cannot switch the limit and the integral, for if we could, we could start by showing that lim n→∞ n cos(√^4 x/n²)/1 + n²x² = 0 for any nonzero x, and then conclude that the answer would be 0. This answer is incorrect, and a ...

Verified Answer:

With the reasoning just presented, we see that it ...

Question: P.172

Let ABCD be a parallelogram in the plane. Describe and sketch the set of all points P in the plane for which there is an ellipse with the property that the points A, B, C, D, and P all lie on the ellipse. ...

Verified Answer:

Solution 1. To show why this description is correc...

Question: P.171

If an insect starts at a random point inside a circular plate of radius R and crawls in a straight line in a random direction until it reaches the edge of the plate, what will be the average distance it travels to the edge? ...

Verified Answer:

The average distance traveled to the edge of the p...

Question: P.170

Suppose we start with a Pythagorean triple (a, b, c) of positive integers, that is, positive integers a, b, c such that a² + b² = c² and which can therefore be used as the side lengths of a right triangle. Show that it is not possible to have another Pythagorean triple (b, c, d) with the same ...

Verified Answer:

Suppose we did have positive integers a, b, c, d s...

Question: P.169

Suppose we are given an m-gon and an n-gon in the plane. Consider their intersection; assume this intersection is itself a polygon. a. If the m-gon and the n-gon are convex, what is the maximal number of sides their intersection can have? b. Is the result from (a) still correct if only one of the ...

Verified Answer:

a. The maximal number of sides that the intersecti...

Question: P.168

a. Find all lines which are tangent to both of the parabolas y = x² and y = −x² + 4x − 4. b. Now suppose f(x) and g(x) are any two quadratic polynomials. Find geometric criteria that determine the number of lines tangent to both of the parabolas y = f(x) and y = g(x). ...

Verified Answer:

a. There are two such lines: y = 0 and y = 4x − 4....

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