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Instantaneous Speed of a Ball In physics it is shown that the height $s$ of a ball thrown straight up with an initial speed of 96 ft/sec from ground level is$$s=s(t)=-16 t^{2}+96 t$$where $t$ is the elapsed time that the ball is in the air.(a) When does the ball strike the ground? That is, how long is the ball in the air?(b) What is the average speed of the ball from $t=0$ to $t=2 ?$(c) What is the instantaneous speed of the ball at time $r ?$(d) What is the instantaneous speed of the ball at $t=2 ?$(e) When is the instantaneous speed of the ball equal to zero?(1) How high is the ball when its instantaneous speed equals zero?(g) What is the instantaneous speed of the ball when it strikes the ground?

(a) 6 sec (b) 64 ft/sec (c) (c) $(-32 t+96) \mathrm{ft} / \mathrm{sec}$ (d) $32 \mathrm{ft} / \mathrm{sec}$ (e) $3 \sec$ (f) $144 \mathrm{ft}$ (g) $-96 \mathrm{ft} / \mathrm{sec}$

Calculus 1 / AB

Chapter 14

A Preview of Calculus: The Limit, Derivative, and Integral of a Function

Section 4

The Tangent Problem; The Derivative

Limits

Differentiation

Integrals

Continuous Functions

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Okay, so report A We want to know when our ball starts the ground. So that's wonderful. And function s entity is equal to go so we can factor out a 16 t from our turn here. Do we get 16 t times a negative T plus 96/16 6 So this is equal to zero. Okay, so we get to solutions when negative t plus six is equal to go and when 16 t is you could video. So this gives us when t is you could to six on when t is equal to zero. Okay. And I'll for part B, we went to find or average speed of our ball from TZ federal to Tuz too. So that's our change And why over our change. So that's gonna be s evaluated. Cuts to minus s evaluated a thorough over a two minus Errol. So this gives us well as evaluated at two. That's a negative 16 times to three part two plus 96 times too. So let's put that into our calculator, and I get 128 and then we have us evaluated at so So that's going to be Joe So we get 128 minus 0/2. So 128 divided by two is going to be equal to 64. And now we want to find our instantaneous rate of change. So that's going to be or derivative. So that's us prime at sea. That's equal to our limits as T approaches some value. See, So we get, um, our function, which is negative 16 t Square plus 96 t And then we're going to subject that by, um, our function evaluated at sea. So that's going to be minus so plus a 16 c squared minus a 96 c all over t minus sea yet. So for party, we want to find all right derivative 20 is equal to two. So it's plug that into our equation. So that's going to be equal to the limits as t approaches to of negative 16 teach squared plus 96 t And then we have, um, play in our Steve for you said that 16 times two to the power of to minus 96 times too. So we have mice 128 over t minus to so Let's fact there are numerator. Okay, so we can practice into a Let's write that down here. The limits as t approaches to of negative 16 times are come T minus four and or a T minus two. So we see that we can cancel out that t minus still and now, using drugs up, we have negative 16 times two minus four. So that's going to be an eight of 16 times native to which is equal to 32.

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