SHOW ALL WORK FOR BRAINLIEST f(t)= -16t^2+64 gives the height of an egg t seconds after being dropped a. make a table using the domain 0_ b. describe where the function is increasing and decreasing in interval notation

SHOW ALL WORK FOR BRAINLIEST
f(t)= -16t^2+64 gives the height of an egg t seconds after being dropped
a. make a table using the domain 0_ b. describe where the function is increasing and decreasing in interval notation

Answer:

the function f(t) = -16t^2 + 64 is increasing for the interval (0, ∞) and decreasing for the interval (-∞, 0).

Step-by-step explanation:

To create a table using the given function f(t) = -16t^2 + 64 for the domain 0 ≤ t, we can substitute different values of t into the equation and calculate the corresponding heights.

Let's create a table with values of t and f(t):

t f(t)

0 64

0.5 48

1 48

1.5 48

2 32

2.5 0

3 -40

3.5 -88

4 -144

4.5 -208

5 -280

Now, let's describe where the function is increasing and decreasing in interval notation.

The given function f(t) = -16t^2 + 64 is a quadratic function with a coefficient of -16 for the t^2 term. Since the coefficient is negative, the parabola opens downwards.

The function will be increasing when the derivative is positive, and decreasing when the derivative is negative.

To find the derivative, we differentiate f(t) with respect to t:

f'(t) = -32t

The derivative f'(t) is a linear function with a coefficient of -32. Since the coefficient is negative, the function f(t) is decreasing.

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The function f(t) = -16t^2 + 64 describes the height of an egg t seconds after being dropped.

a. To make a table using the domain 0 ≤ t ≤ 2, we can plug in values for t and calculate the corresponding values for f(t) as shown in the image I provided.

b. The function is decreasing on the interval [0, ∞) because the leading coefficient of the quadratic term is negative. This means that as t increases, the value of f(t) decreases.

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