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# Prove or disprove given statements for functions such that f(x) = o(g(x))

Let and be functions, both differentiable in a neighborhood of 0, with and such that

Prove or disprove each the following statements.

1. as .
2. as .

1. True.
Proof. Since as we know by the definition of that

Thus, for every there exists a such that

So, for we have

The final line follows since by hypothesis. Therefore,

Hence,

By definition, we then have

2. False.
Consider for and for . Then, for ,

For we have .

Next,

Since we have as . However, since

does not exist.

# Prove an inequality of exponentials

For all and for any constants such that prove that

Proof. We want to consider the function

If we can show this function is decreasing on the positive real axis then we establish the inequality since this would mean that if then

(So, the trick here is to think of this as a function of the exponent. The and are some positive fixed constants.) To take the derivative of we use logarithmic differentiation,

Multiplying both sides by we then obtain

Now we can conclude that for all since the first term in the product

Since (any real power of a positive number is still positive) and . For the second term we have

since and are positive, but both logarithms are negative. We know these logarithms are negative since

implies

Hence, for all . This means is a decreasing function. Therefore, if then we have

# Prove some inequalities of sin x

For all prove that

From the first exercise of this section on inequalities, we know for all . But since for all and we have the inequality on the right immediately,

For the inequality on the left let

Then, we’ll consider the first two derivatives of to show that it is positive for all .

We know from the inequality on the right that for all . Hence, for all . Therefore, is increasing on the positive real axis. Since

we then have that for all . Hence, is increasing on the positive real axis, and since

we have that for all . Thus, for all ,

# Prove the inequality 2x/π < sin x < x for 0 < x < π/2

Prove the inequality

Proof. Define a function . Then, since for we have

Since and is increasing on (since its derivative is positive) we have

For the inequality on the left (which is much more subtle), we want to show

So, we consider the function

We know

and

Now, if we can show that is decreasing on the whole interval then we will be done (since this would mean on the whole interval since if it were less than somewhere then it would have to increase to get back to on the right end of the interval).

To show is decreasing on the whole interval we will show that its derivative is negative. To that end, define a function

(This is the numerator in the expression we got for . Since we know the denominator of that expression is always positive, we are going to show this is always negative to conclude is always negative.) Now, and

for . Thus, is negative, and so is decreasing. Since we then have is negative on the whole interval. Therefore, on the interval. Hence, is decreasing. Hence, we indeed have

for all

# Prove a formula for the 2nth derivative of x sin (ax)

Define and prove the formula for the th derivative of ,

Proof. The proof is by induction. For the case we need to take two derivatives of (since ).

So, the formula holds for the case . Assume then that the formula is true for some positive integer . Then the inductive hypothesis is that,

Now, we want to take two derivatives (to get a formula for ).

Now, we take another derivative,

Thus, the formula holds for . Hence, it holds for all positive integers

# Prove the formulas for derivatives of products and quotients

Derive the formulas for the derivative of a product and the derivative of a quotient from the corresponding formulas for the derivative of a sum and the derivative of a difference.

We know the derivative rules for sums and differences are:

To derive the derivative rule for products using logarithmic differentiation we let and compute

This is the usual rule for derivative of a product.

Similarly, for the derivative of a quotient, let and then compute,

Which is the usual rule for derivative of a quotient.

# Prove that a function satisfying given properties must be ex

Given a function satisfying the properties:

and

Prove the following:

1. The derivative exists for all .
2. We must have .

This problem is quite similar to two previous exercises here and here (Section 6.17, Exercises #39 and #40).

1. Proof. To show that the derivative exists for all we must show that the limit

exists for all . Using the given properties of we can evaluate this limit

Therefore, for all , so the derivative is defined everywhere

2. Proof. From part (a) we know . By Section 6.17, Exercise #39 (linked above) we know that the only functions which satisfy this equation are for all or for some constant (where in the linked exercise). However, since the derivative of exists everywhere, and differentiability implies continuity, we know is continuous everywhere. Hence, . Then,

since , so . Therefore, we must have for some constant . Furthermore, we must have since . Thus,

# Find the slope and area under the graph for a given function

Let

1. Determine the slope of the graph of at the point with -coordinate 1.
2. Find the volume of the solid of revolution formed by rotating the region between the graph of and the interval about the -axis.

1. To take this derivative, using logarithmic differentiation will be easier,

Then differentiating both sides we have,

So, to find the slope at the point with we evaluate,

2. First, the integral to compute the volume of the solid of revolution is,

To evaluate this we use the partial fraction decomposition,

This gives us the equation

Evaluating at , , and we obtain

Therefore, we have

# Prove that x – (1/3)x3 < arctan x if x is positive

Consider the function

Compute the derivative and use the sign of the derivative to prove the inequality

Proof. First, we compute the derivative

Therefore, for all . This implies is strictly increasing if , and in particular, is increasing for . This implies

# Prove a property of the derivative if arctangent and the logarithm obey a given relation

If

prove that

Proof. First, we consider the derivatives of the left and right side of the given equation. (Treating as a function of and remembering to use the chain rule.) So, for the derivative on the left, we have

On the right we have,

Now, using the given equation we have