A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 33 ft/s. Its height in feet after t seconds is given by y=33t-23t^2.

Question

A. Find the average velocity for the time period beginning when t=1 and lasting: .01s, .005s, .002s, .001s. NOTE: For the above answers, you may have to enter 6 or 7 significant digits

Answer 1

Average velocity = (change in height) / (change in time)

So, [Average velocity from t = 1 to t = 1.01] = [33(1.01) - 23(1.01)² - (33(1) - 23(1)²)] / (1.01 - 1)

Do the others in a similar manner.

Answer 2

Since this is listed as a Calculus question, we need to consider the key words in the question, and the concepts you have learned, aka formulas, that can help with this process.

There are two key words in this question: Velocity and Average. Let's start with Velocity: How is velocity related to height in calculus terms?

Velocity_y is the derivative of the height function. (I put the subscript because the direction does matter, and this is a function of height, set equal to y.)

Another word for average (and they will always use the word average in these questions...) is Mean.

If you want to calculate the average velocity, the derivative of the height function, you can use the Mean Value Theorem:

f'(c) = f(b) - f(a) / (b - a)

so inserting the formula:

f'(c) = ([33b - 23b²] - [33a - 23a²]) / (b - a)

where a = 1, and b = 1 + Δ, where Δ is each of the values provided, like 0.01, etc.

Calculus teachers and AP exams really like to only give credit if you show that you are applying a calculus concept, and not just solving through algebraically.