Complex Numbers Quiz

Complex Numbers Quiz

1. Use the method of approximation using derivatives to estimate the

value of √62 .

2. A grandfather clock has a pendulum whose motion has a period of

P=2π

3.13 √L seconds, where L is the length of the pendulum.

(i) Show that when the pendulum has length L = 1 metre, P ≈2.007 seconds.

(ii) Use calculus to find a linear model that approximates the period when the length of the pendulum is near 1 m.

(iii) What is the approximate error in the period if the length of the pendulum was measured incorrectly as 0.9 meters.

3. Using the method of approximation by derivatives, which of the following is the best approximation for 10022?

A 10002

B 10002 + 4C 10002 + 2 4D 10002 + 2 1000E 10002 + 4 1000

4. Suppose the Earth were a perfect sphere and we measured its radius to be

6731 km ± 0.1 km.

Estimate the approximate error in our calculation of the Earth’s surface area.

5. According to the ‘perfect gas law’ the pressure exerted by a fixed amount of gas at a fixed temperature is inversely proportionalto the volume of the container, where K is a constant.

P=KV

For example, the pressure exerted by 1.25g of nitrogen at 20°C in a flask of volume, V litres

is P=108.65

V kilopascals.

Suppose that the flask has an adjustable lid and that the volume is increased from 0.3L to 0.32L. Use calculus to estimate the change inpressure of the nitrogen.

1. Use the method of approximation using derivatives to estimate thevalue of √62.

2. Using the method of approximation by derivatives, which of the following is the best approximation for 10022?

F 10002

G 10002 + 4H 10002 + 2 4I 10002 + 2 1000J 10002 + 4 1000

3. A grandfather clock has a pendulum whose motion has a period of

P=2π

3.13 √L seconds, where L is the length of the pendulum.

(i) Show that when the pendulum has length L = 1 metre, P ≈2.007 seconds.

(ii) Use calculus to find a linear model that approximates the period when the length of the pendulum is near 1 m.

(iii) What is the approximate error in the period if the length of the pendulum was measured incorrectly as 0.9 meters.

4. Suppose the Earth were a perfect sphere and we measured its radius to be

6731 km ± 0.1 km.

Estimate the approximate error in our calculation of the Earth’s surface area.

5. According to the ‘perfect gas law’ the pressure exerted by a fixed amount of gas at a fixed temperature is inversely proportionalto the volume of the container, where K is a constant.

P=KV

For example, the pressure exerted by 1.25g of nitrogen at 20°C in a flask of volume, V litres

is P=108.65

V kilopascals.

Suppose that the flask has an adjustable lid and that the volume is increased from 0.3L to 0.32L. Use calculus to estimate the change inpressure of the nitrogen.

This depends on the addition formulas for the sine and cosine:

sin(x + y) = sin(x) cos(y) + sin(y) cos(x)

cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

DeMoivre's theorem states that:

(cos(x) + i sin(x))^n = cos(nx) + i sin(nx)

In your case, you are assuming this holds for n = k and are trying to prove that it holds for n = k + 1, so you are looking at:

(cos(x) + i sin(x)) ^(k + 1)

This is just:

(cos(x) + i sin(x))^k * (cos(x) + i sin(x))

Using the induction hypothesis, this is:

(cos(kx) + i sin(kx)) ( cos(x) + i sin(x))

Multiplying out and using the fact that i^2 = -1, we get:

cos(kx) cos(x) - sin(kx) sin(x) + i (sin(kx) cos(x) + cos(kx) sin(x))

This is just cos((k+1) x) + i sin((k+1) x) by the addition formulasmentioned above.

I recommend that you study this proof carefully, as it is very typical of induction proofs.

We consider three cases.

For n > 0, we proceed by mathematical induction. When n = 1, the result is clearly true. For our hypothesis, we assume the result is true for some positive integer k. That is, we assume

Now, considering the case n = k + 1:

We deduce that the result is true for n = k + 1 when it is true for n = k. By the principle of mathematical induction it follows that the result is true for all positive integers n ≥ 1.

When n = 0 the formula is true since cos (0x) + i sin(0x) = 1 +i 0 = 1.

When n < 0, we consider a positive integer m such that n = −m. So

Hence, the theorem is true for all integer values of n.