1. ## Velocity and Acceleration--Derivatives

Which graph has:

(a) Constant velocity

(b) Greatest initial velocity

(c) Greatest average velocity

(d) Zero average velocity

(e) Zero acceleration

(f) Positive acceleration throughout

Thank you =)

3. Hello, melliep!

Recall that: .slope = velocity.
. . . . . . .concavity = acceleration

Code:
    (1) |                     (2)  |          *
|                          |
|   *                      |         *
|*     *                   |        *
- * - - - * - - - * -      - + - - -*- - -
|        *     *           |   *
|           *              *
|                          |

(3) |          *           (4) |
|      *                   *
|   *                      | *
| *                        |   *
|*                         |     *
|                          |       *
- * - - - - - - -          - + - - - - * -
|                          |

Which graph has:

(a) Constant velocity

Constant velocity = constant slope.

If the slope doesn't change, we have a straight line . . . graph (4)

(b) Greatest initial velocity

The graph with the steepest slope at $\displaystyle t = 0$ is graph (3).

(c) Greatest average velocity

Graph (1) has a slope that is positive sometimes, negative sometimes.
I suspect that the average velocity is 0.

Graph (4) has a constant negative slope.

Graph (3) always has a positive slope, but it is decreasing.

Graph (2) always has a positive slope and it is increasing.

I'd bet on Graph (2).

(d) Zero average velocity

As I suspected, it must be graph (1).

(e) Zero acceleration

Zero acceleration = zero concavity.

We must have a straight line . . . graph (4).

(f) Positive acceleration throughout

Positive accelration = positive concavity (concave up).

The curve that is always concave up is graph (2).

4. you are utterly amazing. thank you so much i really really appreciate it.