Use induction to prove the following inequality for 1 ≤q n and

Problem 1: Use induction to prove the following inequality for $1 \leq n$ and $-1<x<0$ and $0<x$

$(1+x)^{n} \geq 1+n x$

Note that the equality holds for $x=0$.

Problem 2: Prove the following inequality

$\tan ^{-1} v-\tan ^{-1} u<v-u \text { if } u<v$

Solution: We know that

\[f\left( x \right)-f\left( y \right)=f'\left( c \right)\left( x-y \right)\]

for some $c\in [x,y]$, assuming that f is continuously differentiable. Let $f\left( x \right)={{\tan }^{-1}}\left( x \right)$, then

\[f'\left( x \right)=\frac{1}{1+{{x}^{2}}}\]

Observe that

\[0\le f'\left( x \right)=\frac{1}{1+{{x}^{2}}}<1,\text{ for all }x\in \mathbb{R}\]

Hence,

\[{{\tan }^{-1}}v-{{\tan }^{-1}}u=f'\left( c \right)\left( v-u \right)<1\cdot \left( v-u \right)=v-u\]

(since $v-u>0$ and $f'\left( c \right)<1$ ).

Problem 3: Assuming that the first derivative test holds, prove the following statement known as the second derivative test. "Suppose $f^{\prime}(a)=0$. If $f^{\prime \prime}(a)>0$, then $f$ has a local minimum at $\mathrm{x}=\mathrm{a}$; if $f^{\prime \prime}(a)<0$, then $\mathrm{f}$ has a local maximum at $\mathrm{x}=\mathrm{a} "$.

Problem 5: A wire of length can be shaped into a closed semicircle or a circle. The wire can also be divided into two pieces, with one piece forming a (closed) semicircle, and the other a circle. Find the largest and smallest areas that are possible.

Problem 6: The following problem concerns with the most economical dimensions of a cylindrical tin can that holds a predetermined volume $\mathrm{V}$. When the circular top and bottom are cut from a large sheet of metal, there will be wastage. Thus the amount of metal necessary to make a can is actually larger than the amount of metal used in the can itself. Suppose that the top and bottom are each cut from square with side lengths that are the diameter of the circle. What is $h / r$ when the amount of metal (including wastage) is minimized?

Problem 7: (use antiderivatives to this problem. Justify your reasoning) As a scientist in the pathfinder mission to Mars, you are presented with the following problem: The acceleration of gravity near the surface of Mars is $4 \mathrm{~m} / \mathrm{s}^{2}$. If an object (the pathfinder in this case) is propelled downward from a height of $15 \mathrm{~m}$ and at a speed of $72 \mathrm{~km} / \mathrm{hr}$, in how many seconds will it hit the surface of Mars. What is the velocity at impact?

Problem 8: ( a problem from business applications see section 4.6)

The fixed cost of producing a compact disc is 12000 dollars. The marginal cost per disc is 1.20 dollars. Market research and previous sales suggest that the demand function for this particular disc will be $x=19000-600 p$. At what production level is the marginal cost equal to marginal revenue (another way of saying this is at what production level the profit is maximized)? Hint : Find the cost function first.

Price: $19.67

Solution: The downloadable solution consists of 11 pages, 867 words and 1 charts.
Deliverable: Word Document