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# Some true/false questions

For each statement, prove that it is true or show that it is false.

1. .
2. .
3. for every .
4. for all .

1. True.
Proof. We can compute using the definition of the exponential

2. False.
On the left we have

While on the right we have,

But since , these two quantities cannot be equal.

3. True.
Proof. The proof is by induction. For the case on the left we have

While on the right we have

Therefore, indeed for the case .

Assume then that the statement is true for some positive integer . Then,

Thus, the inequality holds for the case ; hence, it holds for all positive integers

4. False.
From the definitions of and we have

Using these definitions, the inequality states

However, this is false if since for .

# Find all x satisfying equations given in terms of sinh

Let be the number such that . Find all that satisfy the given equations.

1. .
2. .

1. We are given . From the formula for this means

Then, from the given equation we have

Thus,

So, then we have

Therefore we have

2. There can be no which satisfy the given equation. As in part (a), we use the definition of to obtain the equation,

Next, we use the equation given in the problem to write,

Furthermore, we can obtain an expression for by considering

Putting these expressions for and into our original equation we have

But this implies

which is impossible. Hence, there can be no real satisfying this equation.

# 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.

# Compute f(x) + f(1/x) where f satisfies a given integral equation

Define a function by an integral equation as follows:

Using this formula compute .

Substituting and into the formula for we have

For the second integral we make the substitution , . This gives us

As a check, we evaluate at ,

# 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

# Find the derivative of log (arccos (x-1/2))

Find the derivative of the function

Using the chain rule and the formulas for the derivative of the logarithm and the arccosine we have,

The formula is valid for .

# Prove formulas for the partial derivatives of xy

Define a function of two variables, with ,

Prove

Proof. First, we write,

Then we compute the derivatives using the chain rule and formulas for the derivatives of the exponential and logarithm,

And,

# Find the derivative of a product of terms (x-ai)bi

Find the derivative of the following function:

To take this derivative we want to use logarithmic differentiation. To that end we take the derivative of both sides,

Therefore, taking the derivative of both sides, we have

# Find the derivative of (x2(3-x)1/3)/((1-x)(3+x)2/3)

Find the derivative of the function

First, we take the logarithm of both sides so we can use logarithmic differentiation,

Then we differentiate both sides,

Therefore,

# Find the derivative of x1/x

Find the derivative of the function

We take the derivative using logarithmic differentiation. First, taking the logarithm of both sides,

Then take the derivative of both sides,