Alan

A secant line is a straight line so it will have an equation of the form y = m*x + c where m and c are constants.

To find the constants first substitute the given values of x into f(x) to find the corresponding values of y

y₃ =  3^3 + 2*3 - 1 or  y₃ =  32

y₅ = 5^3 + 2*5 -1  or  y₅ = 134

Now use these values in the equation for the straight line

32 = m*3 + c                 (1)

134 = m*5 + c               (2)

Subtract (1) from (2)

102 = 2m so m = 51

Put this back into (1)

32 = 51*3 + c

32 = 153 + c

c = 32 - 153 or c = -121

So the equation of the secant line is y = 51x - 121

I'll do the first one for you:

x = t^2 + 5t + 2

v = Limit_dt→0 (x_t+dt - x_t)/dt             I'll use dt instead of writing out deltat all the time.

so

Put t+dt into the expression for x to get the x_t+dt part

v = Limit_dt→0  ([t+dt]^2 + 5[t+dt] + 2 - t^2 - 5t - 2])/dt 

Expand the terms in brackets

v = Limit_dt→0  (t^2 + 2tdt + dt^2 + 5t + 5dt + 2 - t^2 - 5t - 2])/dt 

Subtract terms where appropriate

v = Limit_dt→0  (2tdt + dt^2 + 5dt])/dt 

Divide numerator by dt

v = Limit_dt→0  (2t + dt + 5] 

Let dt go to zero and you are left with:

v = 2t + 5 

See if you can apply a similar approach with the others.

Velocity is rate of change of distance, so, if x is distance, velocity,v is given by:

l.   2t + 5

ll.  1/2

lll.  t/2

lV. 2t + 3/4

Just plug in t = 5 (where approriate) to get the velocities.

The ratio of the circumference to the diameter of a circle.

b.  tan(elevation angle) = (height of flagpole above ground - height of student above ground)/distance from base

tan(22°) = (h - 18)/50   where h = height of flagpole above ground

Multiply both sides by 50

50*tan(22°) = h - 18

Add 18 to both sides 

h = 50*tan(22°) + 18  ≈ 38.2 ft

d. 

Natural numbers are the counting numbers 1, 2, 3, 4, ... etc.

tan(elevation angle) = height of pole/length of shadow

elevation angle = tan-1(4/6) ≈ 33.7°

Yes. You can simply type it as you have written it, with the number or expression of interest in the brackets

This can be written as 16(x + n/4)^2 = 0 which has the solution x = -n/4. Don't forget the negative sign!