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# Prove that a function satisfying a given equation must be the exponential function

If is a function defined for all and such that

prove that we must have

for some constant .

Proof. Let . Then we can compute the derivative,

Since we know we then have

Therefore, for some constant and we have

# Deduce a limit relation using the derivative of ecx

Consider the function for a constant . First, show that and then prove

Proof. If then we have

From the definition of the derivative we also know

# Show that a given function satisfies a functional equation

Define a function for by

Prove that

Proof. First, we compute and using the definition of above, and the properties of the exponential established in the previous exercise (Section 6.17, Exercise #36), we have

Therefore,

# Prove basic properties of the exponential function

A real number raised to a real exponent is defined by

Prove the following properties:

1. .
2. .
3. .
4. .
5. For , if and only if .

For all of these we use the definition and then use the corresponding properties of the exponential function . (These are proved for the function in Theorem 6.6, and then we define in Section 6.14.)

1. Proof.

2. Proof.

3. Proof.

4. Proof.

5. Proof. Assume . For the forward direction, assume . We have

Therefore, since we have , and so

Conversely, if then we have

# Find the derivative of (sin x)cos x + (cos x)sin x

Find the derivative of the function

Rewrite each of the exponentials in the expression for using the definition of the exponential function,

Now, we take the derivative directly using the chain rule and the formula for the derivative of the exponential function

# Find the derivative of log (ex + (1+e2x)1/2)

Find the derivative of the following function:

We compute using the chain rule and the formulas for derivatives of logarithms and exponentials,

# Find the derivative of (ex – e-x)/(ex + e-x)

Find the derivative of the following function:

We can compute this derivative directly using the quotient rule,

# Find the derivative of (1+x)(1+ex2)

Find the derivative of the following function:

We can compute this derivative by multiplying out the expression for and using the product and chain rules. First, we multiply out the expression,

Then, we take the derivative,

# Prove integration formulas for eaxcos (bx) and eaxsin (bx)

Let and be constants with at least one of them nonzero and define

Using integration by parts, establish the following formulas for constants ,

Using these formulas prove the following integration formulas,

To establish the formula we use integration by parts letting

Then we can evaluate using the formula for integration by parts,

To establish the second formula, , we use integration by parts again. Let

Then we have

This establishes the two requested equations, now we prove the two integral identities.

Proof. Solving for in the second equation above we have

Plugging this into the first equation we have

Next, for the second integral equation we are asked to prove, we use the formula we obtained for above,

Then, we use the expression we obtained for into this,

This implies,

# Find constants so that ex satisfies a given integral equation

Find values for the constants and such that the following equation is true:

To find values of the constants we evaluate the integral,

Therefore, we need to find constants such that

Therefore, we can choose any value for and then set and this pair will make the equation true for all .