Chapter 8 - Differentiation and integration

Question 1

Define “limit” (limit of functions)

Answer 1

A function f(x) is said to approach a limit L as x approaches the value a if, given any small positive quantity ß, it is possible to find a positive number K such that |f(x) - 1| < ß for all x satifying 0 < |x - a| < K.

Less formally, this means that we can make the value of f(x) as close as we please to L by taking x sufficiently close to a. Note that, using the formal defnition, there is no need to evaluate f(a); indeed, f(a) may or may not equal L. The limiting value of f as x -> a depends only on nearby values!

Question 2

What is a continuous function?

Answer 2

If the function f is defined for a and and for values near a, the function f is continuous for x = a and the two equations (picture) are satisfied.

Question 3

What is the definition of the derivative of the function f(x) at the point x?

Answer 3

Question 4

What are the conditions for f ‘ (x) to exist in the interval [a, b]?

Answer 4

The function f must be “well behaved”, meaning it has to be continuous, it can’t be vertical, and have no sharp corners.

Question 5

Define the constant multiplication rule for differentiation.

Answer 5

Question 6

Define the sum rule of differentiation.

Answer 6

Question 7

Define the product rule of differentiation.

Answer 7

Question 8

Define the quotient rule of differentiation.

Answer 8

Question 9

(ax + b) ‘ = ___

Answer 9

(ax + b) ‘ = a

Question 10

If f(x) = C, where C is a constant, what is f ‘ (x)?

Answer 10

f ‘ (x) = 0

Question 11

For all real numbers n, what is (xⁿ)’ ?

Answer 11

(xⁿ)’ = n * x^n-1

Question 12

d/dx (u(x) + v(x)) = (u(x) + v(x)) ‘ = ____

Answer 12

(u(x) + v(x)) ‘ = u’(x) + v’(x)

Question 13

Solve:

(x²+x³) ‘ = ___

Answer 13

= (x²) ‘ + (x³) ‘ = 2x + 3x²

Question 14

Using the quotient rule (or the exponent rule), what is (1/x) ‘ ?

Answer 14

(1/x) ‘ = - (1/x²)

Question 15

Answer 15

Question 16

Define the chain rule in differentiation.

Answer 16

Question 17

Use the chain rule to differentiate y = (3x + 1)²

Answer 17

y ‘ = (u(x)) ‘ * u(x) ‘

= 2(3x + 1) * 3

= 6(3x + 1)

= 18x + 6

Question 18

Solve the equation (picture) using the chain rule.

Answer 18

Question 19

tan x ‘ ?

Answer 19

Question 20

d/dx (sin^-1x) = (sin^-1x) ‘ = ___

Answer 20

Question 21

d/dx (cos^-1x) = (cos^-1x) ‘ = ___

Answer 21

Question 22

d/dx (tan^-1x) = (tan^-1x) ‘ = ___

Answer 22

Question 23

Answer 23

Question 24

Answer 24

Question 25

Differentiate the function (picture)

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Question 29

Answer 29

Question 30

Differentiate the function (picture)

Answer 30

Question 31

What does the picture mean?

Answer 31

The second derivative.

= f ‘ ‘ (x)

Question 32

How do you calculate the volume of a solid of revolution (picture)?

Answer 32

Question 33

How do you find the centre of gravity of a solid of revolution?

Answer 33

By symmetry, the centre of gravity lies on the x axis, so that the y-coordinate is zero

Y = 0

We find the x-coordinate by the formula (picture):

Question 34

How do you calculate the mean value of a function in a given interval?

Answer 34

Question 35

Given a function y, how do you calculate the length of the curve in a given interval?

Answer 35

Question 36

How do you solve the equation (picture):

Answer 36