1. On the right side of the equation below, we seek an expression for how long it would require the antto take a short lenfth h of soil and then carry it a distance x to the mouth of the tunnel.T(x+h)-T(x)Notice that T(x + h) − T(x) ‡ T(h), for T(h) represents the time it takes to extend the tunnel adistance h from the mouth of the tunnel (at x = 0), while T(x + h) - T(x) includes this time plusthe time it takes to carry the soil to the mouth of the tunnel.=Below are a few possibilities for the right side of the above equation. Defend or reject each and offeryour reasons. One idea to consider in trying to reject a model consider some extreme cases and seeif the model makes sense, e.g., h = 0 or x = 0, or either h or x very large. You might have otherreasons.(a) T(x+h) — T(x) = x+h(b) T(x+h) – T(x) = x - h(c) T(x + h) − T(x) = xh(d) T(x+h) - T(x) = x * h(e) T(x+h) − T(x) = hª(f) T(x+h)-T(x) = c Ants are extremely industrious and tireless tunnel builders. How long does it take an ant to build a singletunnel? To address the issue we need to narrow the scope of the problem, simplify, and identify some termsand parameters.To begin, let's define a few crucial variables. Consider a single ant digging a tunnel into a hillside asillustrated in Figure 1. Let x denote the current length, in feet, of the tunnel that the ant is digging. LetT(x) be the time, in hours, it has taken the ant to build the tunnel of length x. This is a rare case in thisclass where the independent variable is not time!TunnelFigure 1: Sketch of ant building a tunnel. Here x is the current length of the tunnel and T(x) is the timeit has taken the ant to build the tunnel of length x.E+hX-Instead of defining T(x) directly, we'll instead consider Figure 2, a depiction of the essential aspects ofthe situation. When an ant digs a tunnel, the ant must extend the tunnel incrementally, by removing soilbetween coordinates x and x + h, carrying this soil back to the tunnel entrance, and then returning toremove more soil. Let us consider how long it takes the ant to complete the extension of the tunnel fromlength x to length x + h.TunnelRemovedSoilFigure 2: A useful diagram for modeling the time it takes to extend a small section of the tunnel fromdistance x to x + h.

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In the given problem, the function that governs the time taken by the ant to dig out a tunnel of…