Consideration of general features of the contextFor k = 0.2;a = 1, sketch the graphs of ƒ1(t) = 5e–kt , ƒ2(t) = –5e–kt and s(t) = 5e–kt sin(at), where t ∈ [0, 4π].Find the derivative of s(t), in terms of t, k and a, and hence for k = 0.2; a = 1, find the coordinates of the first two maximum/minimum points for s(t) with x coordinates closest to the y-axis.

To solve this problem, we need to follow these steps:

Step 1: Sketch the graphs of ƒ1(t) = 5e–kt , ƒ2(t) = –5e–kt and s(t) = 5e–kt sin(at), where t ∈ [0, 4π].

For ƒ1(t) = 5e–kt and ƒ2(t) = –5e–kt, these are exponential decay functions. The graph of ƒ1(t) starts at (0,5) and approaches the x-axis as t increases, while the graph of ƒ2(t) starts at (0,-5) and approaches the x-axis as t increases.

For s(t) = 5e–kt sin(at), this is a damped sine wave. The amplitude of the sine wave decreases as t increases due to the exponential decay factor 5e–kt.

Step 2: Find the derivative of s(t), in terms of t, k and a.

The derivative of s(t) can be found using the product rule for differentiation, which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first.

So, the derivative of s(t) is:

s'(t) = d/dt [5e–kt sin(at)] = 5e–kt cos(at) * a - 5e–kt sin(at) * k = 5a e–kt cos(at) - 5k e–kt sin(at)

Step 3: For k = 0.2; a = 1, find the coordinates of the first two maximum/minimum points for s(t) with x coordinates closest to the y-axis.

The maximum and minimum points of s(t) occur where the derivative s'(t) = 0.

Setting s'(t) = 0 gives:

5a e–kt cos(at) - 5k e–kt sin(at) = 0

Solving this equation for t will give the x-coordinates of the maximum and minimum points. The corresponding y-coordinates can be found by substituting these x-values into the original function s(t).

Note: The actual solution to the equation s'(t) = 0 will depend on the specific values of k and a, and may require numerical methods to solve if it cannot be solved analytically.