Week 2

Question 1

What is the form of an exponential function?

Answer 1

f: R -> R; f(x) = b * a^x

for some real numbers a > 0, b =/ 0

b = initial value
a = base
x = exponent

Question 2

What is euler’s number?

Answer 2

e = 2.71828 (approx.)

Question 3

How to approximate euler’s number?

Answer 3

One can approximate e by (1 + n1 )n for large n

Question 4

What is exponential growth and decay?

Answer 4

If a > 1 = growth

If 0 < a < 1 = decay

Question 5

Properties of exponential functions?

Answer 5

Zeroes/Roots = No zeroes, either always positive or negative if b is pos or neg, respectively.

Bijection = Cannot be injective if f(1) as this means constant, NOT one-to-one. Not bijective.
Injective IF a =/= 1, injective

Not surjective if codomain is R
Surjective if codomain interval is (0, infinity) if 0 < b
Surjective if codomain interval is (-infinity, 0) if 0 > b

Convex/Concave = Convex if b > 0, Concave if b < 0

Question 6

Properties of logarithmic functions?

Answer 6

Zeroes / Roots = Have zero at x = 1 (loga(1) = 0 = a^0 = 1)

Bijection = Since it is inverse of exponential (reflection of exp. function along line x = y, always bijective
f: (0, infinity) -> R

Cocave/Covex = Concave if a > 1, Convex if a < 1

Question 7

One way to determine if data set is exponential?

Answer 7

Convert y value to log form. If resulting graph is a line, then original is exponential.

Question 8

Do exponential decay functions have the same decay rate as power-law functions?

Answer 8

No, exponential functions decay faster than power-law.

Question 9

What transformation causes power-law functions to appear linear?

Answer 9

Log-log plot transformations

Question 10

What transformation causes exponentials to appear as linear?

Answer 10

Log-lin plots

Question 11

What transformation causes logarithmic functions to appear linear?

Answer 11

Lin-log plots

Question 12

What is ln(x)?

Answer 12

Same as loge (x)

Question 13

What are the reasons predictions can not go well?

Answer 13

1. Accuracy of data
2. Could be a different function that fits data better
3. Real-life constraints can affect predicted data (e.g. bacteria cannot grow infinitely due to limits to space and sustainance)
4. Choice of data points to determine model could be unrepresentative of data / bad fitting straight line.

Question 14

What do derivatives determine?

Answer 14

• Finding minimal or maximal values

- Find parameters that minimise the ‘error’ in a function fitted to given data.

Question 15

What are tangents?

Answer 15

A line at a point that best approximates f(x) for x near the point a.

touches plot in (a, f(a))

Question 16

What are the basic derivative rules?

Answer 16

Constant: If f (x) = c, then f 0(x) = 0.

Polynomial: If f (x) = x^b, then f 0(x) = bx^b-1.

Exponential: If f (x) = a^x , then f 0(x) = ln(a)a^x .

Logarithmic: If f (x) = loga(x), then f 0(x) = 1/ln(a)x

Question 17

How to determine slope of tangent at pont a?

Answer 17

1. Find derivative
2. Sub in a for x
3. Answer is slope of tangent

Question 18

What is the product rule?

Answer 18

derivative of f(x)*g(x) = f’(x)g(x) + f(x)g’(x)

Question 19

What is the chain rule?

Answer 19

The derivative of f(g(x)) = g’(f’(g(x))