The vertical velocity, v(t)v(t)v(t) in metres per second, of a research probe is modelled by v(t)=5tv(t) = 5^tv(t)=5t for t∈[−1,1]t \in [-1, 1]t∈[−1,1], where t t\,t is the time in seconds.
(i) Given that y=5xy = 5^xy=5x, determine an expression for dydx\displaystyle \frac{dy}{dx}dxdy.
(ii) Hence, find ∫5x dx\int 5^x \, dx∫5xdx.
The total distance travelled by the probe, DDD, which is the area bounded by the curve v(t)=5tv(t) = 5^tv(t)=5t, the ttt-axis, and the lines t=−1t = -1t=−1 and t=1t = 1t=1, is approximated using four rectangles of equal width. The rectangles are constructed such that they lie entirely below the curve, as shown in the conceptual Riemann sum model.
(i) Show that the exact area of the largest rectangle is 52\displaystyle \frac{\sqrt{5}}{2}25.
(ii) The areas of these four rectangles form a geometric sequence. Find the exact value of the total area of the four rectangles. Give your answer in the form k(1+5)k(1 + \sqrt{5})k(1+5) where k k\,k is a rational number.
(iii) Find the exact value of the limit of the approximations for D D\,D as the number of rectangles, nnn, increases to infinity (n→∞n \to \inftyn→∞).
