Explain the difference between the average rate of change of y as x changes from a to b, and the instantaneous rate of change of y at x

The average rate of change of a function on the interval [a,b] is the slope of the secant line through (b(,f(b) and (a,f(a)), and is dependent upon the values of a and b specified.

The instantaneous rate of change of that function on the same interval is the LIMIT of this average rate of change of the function as h becomes smaller and smaller. This limit is called the "derivative" of the function f(x).

sec ф = 5/3 (quadrant 4)

let x = base

y = perp

z = hypotenuse

sec ф = hypotenuse/base = 5/3

⇒ z = 5 and x = 3

finding y:

z² = x² + y²

5² = 3² + y²

y² = 25-9

y² = 16

y = 4

now finding all other geometric functions of theta:

sin ф = y/z = 4/5

sin ф = -4/5 (sin is negative in the 4th quadrant)

cos ф = x/z = 3/5

cos ф = 3/5 (cos is positive in 4th quadrant)

tan ф = y/x = 4/3

tan ф = -4/3 (tan is negative in 4th quadrant)

cosec ф = z/y = 5/4

cosec ф = -5/4 (cosec is negative in the 4th quadrant)

cot ф = x/y = 3/4

tan ф = -3/4 (tan is negative in 4th quadrant)

1/4 because there is 4 grades.