Question

Find the exact location of all the relative and absolute extrema of the g(t) = ef - t with domain (-1, 1]

A g has a relative maximum a t (t, y) = __________
B g has an absolute minimum a t (t, y) =__________
C g has an absolute maximum V a t (t, y) =_________

Answer 1

The function g(t) = e^f - t has no relative extrema. The absolute minimum is at t = 1, the endpoint of the domain, yielding the point (1, e^f - 1). There is no absolute maximum within the provided domain.

Step-by-step explanation:

To find extrema in any function, we must first derive it to find its critical points. For the given function g(t) = ef - t, we should first note that ef is a constant since f is not specified as a function of t. As such, the derivative of the function with respect to t is simply -1. Since the derivative does not change, there are no critical points, and hence the function has no relative extrema within the domain (-1, 1].

Next, to find absolute extrema, we evaluate the endpoints of the domain that the function is defined on. At t = -1, g(t) = ef - (-1) = ef + 1, and at t = 1, g(t) = ef - 1. The endpoint t = -1 is not included in the domain, so we only consider t = 1. Since the function is continuously decreasing, the absolute maximum is located at t = -1 if it were included, but this is not part of the domain, so there is no absolute maximum within the given domain. The absolute minimum within the domain is at t = 1, yielding the point (1, ef - 1).

Answer 2

The question involves understanding the relation between position-time graphs and the derived velocity-time and acceleration-time graphs, and identifying times of greatest velocity, zero velocity, and negative velocity or acceleration.

Step-by-step explanation:

The question relates to analyzing graphs of motion, specifically position versus time graphs and their derivatives like velocity versus time and acceleration versus time graphs. By looking at a graph of position versus time, it is possible to infer the changes in velocity and acceleration over time. The gradient of the position-time graph gives the instantaneous velocity: where the gradient is greatest, the instantaneous velocity is highest; where the gradient is zero, the velocity is zero; and where the gradient is negative, the velocity is negative. Similarly, the gradient of the velocity-time graph gives the acceleration: where this gradient is greatest, acceleration is the highest; when the velocity-time curve is flat (gradient zero), acceleration is zero; and where the gradient is negative, the acceleration is negative too.