answer:The simplified form of the expression is frac{-cos left(frac{9}{15}right)-i sin left(frac{9}{15}right)}{4^9}, which can be further simplified to frac{-cos left(frac{3}{5}right)-i sin left(frac{3}{5}right)}{262144}, as cos left(frac{9}{15}right) = cos left(frac{3}{5}right) and sin left(frac{9}{15}right) = sin left(frac{3}{5}right), and 4^9 = 262144.

answer:The width of each subinterval is {eq}Delta x = frac{4-3}{3} = frac{1}{3}{/eq}. The left endpoints are {eq}x_0 = 3, x_1 = frac{10}{3}, x_2 = frac{11}{3}{/eq}. The left Riemann sum is: {eq}begin{align*} L_3 &= sum_{i=0}^{2} f(x_i) Delta x &= Delta x left[ f(3) + fleft(frac{10}{3}right) + fleft(frac{11}{3}right) right] &= frac{1}{3} left[ (16 - 3^2) + left(16 - left(frac{10}{3}right)^2right) + left(16 - left(frac{11}{3}right)^2right) right] &= boxed{3.7037} end{align*}{/eq}

answer:The summation of a sequence of functions T_k = O(n), where 1 leq k leq n, does not necessarily result in a linear sum. While each individual term T_k is O(n), the sum of these terms can lead to a different growth rate. To analyze the sum, consider that T_n = O(n) implies there exists a constant C such that T_n leq Cn. Thus, sum_{k=1}^n T_k leq C sum_{k=1}^n k = C cdot frac{n(n+1)}{2} = O(n^2). This shows that the sum is O(n^2), not O(n). The bound O(n^2) is tighter because it reflects the combined effect of adding n terms, each of which is O(n). If T_n were to be Omega(n) as well, it can be shown that the sum is indeed Omega(n^2), indicating that the O(n^2) bound is the best possible in this case. For example, if T_n = n, the sum would be exactly n^2/2, confirming the O(n^2) behavior.

answer:Fossils offer a wealth of information about the past. They provide evidence of the existence of ancient organisms, their physical characteristics, and their behavior. For example, the discovery of dinosaur fossils has shed light on the diversity and size of these prehistoric creatures. Fossils also help scientists understand how species have evolved over time. By comparing the fossils of different organisms, scientists can trace the changes that have occurred in their anatomy, physiology, and behavior. Fossils also provide insights into past climates and environments. By studying the types of plants and animals found in fossil deposits, scientists can reconstruct ancient ecosystems and determine the conditions in which these organisms lived. For example, the presence of certain types of fossils in a particular location may indicate that the area was once a tropical rainforest or a shallow sea. Furthermore, fossils help us understand the relationships between different organisms. By examining the fossils of organisms that lived together, scientists can infer how they interacted with each other and their environment. For instance, the discovery of fossilized footprints alongside dinosaur bones can provide evidence of predator-prey relationships or social behavior. In summary, fossils provide humans with a wealth of information about the history of life on Earth, the evolution of species, past climates and environments, and the relationships between different organisms. They are invaluable tools for scientific research and contribute to our understanding of the natural world. Fossils provide valuable insights into the history of life on Earth and help us understand the evolution of species. They serve as evidence of past climates, environments, and the relationships between different organisms. By studying fossils, scientists can reconstruct ancient ecosystems and gain knowledge about the origins and diversity of life.