calculus 3
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fntre12345
Science
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hello there is 5 problems in the file inserted below please do all of the hand writting showing all work thank you
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Calculus III
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Quiz
3-8-2018
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--------------------------------------------------------------------------------------------1. Graph at least 5 level curves for each of the following functions.
a) 𝑧 = √𝑥 2 + 4𝑦 2
b) 𝑧 = cos(2𝑥 + 𝑦)
2. Evaluate the following limits.
a)
b)
c)
lim
𝑥𝑦
(𝑥,𝑦)→(0,0) 𝑥 2 +𝑦 2
√𝑥+𝑦−3
(𝑥,𝑦)→(4,5) 𝑥+𝑦−9
lim
lim
𝑥 2 +𝑦 2
(𝑥,𝑦)→(0,0) 𝑥(𝑦 2 −1)
3. The volume of an ellipsoid with axes of length 2𝑎, 2𝑏, 𝑎𝑛𝑑 2𝑐 is 𝑉 =
𝑑𝑣
𝜋 𝑎𝑏𝑐. Find the percentage of the relative change % when 𝑎 increases
𝑣
by 2%, and 𝑏 decreases by 2.5%, and 𝑐 increases by 3%.
4. Find the total differential for 𝑓(𝑢, 𝑣) = 1 − 𝑡𝑎𝑛−1 (𝑢2 𝑣 + 𝑣 2 𝑢).
5. Find 𝐹𝑥𝑦𝑧 , 𝐹𝑦𝑥𝑧 , and 𝐹𝑧𝑥𝑦 for the given function.
𝐹(𝑥, 𝑦, 𝑧) = tan(𝑥𝑦𝑧)
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"Automatically" is a good choice as it leaves a ghost of the image visible until you hover over it.Load the DynamicEarth.kmz file from into Google Earth Pro. It is located at https://serc.carleton.edu/sp/library/google_earth/examples/49004.html (Links to an external site.) and is the top file in the "Description and Teaching Materials" list. You should be able to double-click on the filename and it will open. Or, you can download the file onto your computer first, and then open it in Google Earth Pro by using File/Open and navigating to the file.Once the DynamicEarth.kmz is loaded, click and drag to move it from "Temporary Places" to "My Places." Then save "My Places" by clicking File/Save/Save My Places. DynamicEarth.kmz will now be available every time you open Google Earth Pro on your computer. When you exit, Google Earth Pro should save "My Places" for the next time. But you should manually save "My Places" whenever you make significant changes to it, as Google Earth Pro does not autosave during a session.You now have an interactive view of the Earth! Take some time to explore the Earth with Google Earth and figure out how the navigation works using the keyboard, your touch pad, your mouse. For example:Zoom in and out, move N, S, E, W, grab and spin the globe, etc. The resolution will change as you zoom. Clicking on the "N" of the navigation compass reorients the view so north is "up." At top left, "search" (and fly to) any place of interest. Zoom in and click on the "street view" icon (orange stick figure under the compass at top right) to explore an area as if you were on footZoom in to see individual buildings, roads, cars, etc. Go 3D - zoom into a significant topographic feature (e.g. Mount Everest, the Grand Canyon, Niagara Falls). Hold the Shift key down and tilt the terrain using the Up/Down arrows to tilt the terrain, and spin the terrain using the Right/Left buttons. Do the same thing for topographic features on the ocean floor. Note that under Tools/Options/3D View you can increase the vertical exaggeration by up to 3x. This is useful to emphasize subtle features, but is pretty scary when you look at the Grand Canyon that way!On the Google Earth tool bar, click the clock-with-an-arrow icon to explore historical imagery in an area of interest (views through time of your favorite city, for example)By clicking and dragging, you can move things that you have found and want to save, from the "Search" menu into "My Places." You can also re-organize "My Places" by adding and deleting items, changing the order of things, making subfolders, etc. Explore the built-in items under the Layers menu at bottom left, and Dynamic Earth layers in your Places menu.Expand and contract the folders and subfolders, turn various items on and off, etc. For example, with the Dynamic Earth/Volcanoes of the World layer displayed, right-clicking on a volcano (double-clicking with a Mac) brings up an information box about it. B. Topographic Patterns Uncheck all of the layers and focus on topographic features of the Earth.Topography of the earth ABOVE sea levelAre mountains randomly distributed on the continents, or do they tend to occur in particular patterns (clusters, linear chains, arcs, etc.)?Find Mt. Everest, the highest point on earth. Zoom in enough to see the summit, then pan your cursor around to locate the highest point (elevations shows up in the status bar at the bottom, as long as View/Status Bar is selected). The elevation of Mt. Everest is how many meters?Topography of the earth BELOW sea levelWe are all relatively familiar with the topography of the Earth's surface above sea level, but less so with the bathymetry of the Earth below sea level. Before this was known, most people assumed that the seafloor was relatively flat and featureless, and personal experience with lakes and rivers suggested that the deepest part would be in the middle. Actual mapping of the sea floor, however, showed some surprises. Such mapping began in the 1930's but accelerated during World War II with the advent of submarine warfare. Princeton Geosciences Professor Harry Hess played a pivotal role; as captain of the USS Cape Johnson he used the ship's echo-sounder to "ping" the seafloor and measure depth as the ship traversed the Pacific Ocean between battles. After the war, this data led him to propose seafloor spreading, a process crucial to the development of the theory of plate tectonics.Modern methods to measure bathymetry include multi-beam echo sounders that map a wide swath of seafloor, and satellite measurement of variations in sea level due to variations in gravitational pull over bathymetric features – sea level is slightly lower over low spots on the sea floor and slightly higher over high spots.On Google Earth, the bathymetry is shown in shades of blue: the darker the blue, the greater the depth. You can get Google Earth Pro to draw topographic profiles by a) using the "Add Path" tool to draw a path across a region of interest; b) saving that path to My Places and c) right-clicking on the path in My Places and choosing "Show Elevation Profile." In order to see a bathymetric profile of the sea floor, (as opposed to a topographic profile on land), there is one more important step to take. In the information box for the path you create, click on "Altitude", and then choose "clamped to the sea floor" instead of "clamped to the ground". Otherwise your profile will simply show you a flat line for the sea surface.Examine the Atlantic Ocean between North/South America and Eurasia/Africa. Note that the deepest part is not the middle; instead, an underwater mountain range runs down the middle of the ocean.Features like this are called mid-ocean ridges or spreading ridges (more on the "spreading" later in this lab). Zoom in enough to see that although the ridge is a topographic high, it also has a valley (the "rift valley") running along the middle of it. In the space below, complete the topographic profile of the Atlantic Ocean floor between South America and Africa. Take a digital photograph of your sketch to including in your lab report.Scan around to see the ocean ridges in the Indian, Pacific and Southern Oceans. If the earth's lowest spots aren't in the middle of the ocean, where are they? Focus on the west coast of South America, and in the space below complete the topographic profile of the Pacific Ocean floor from South America westward about 600 miles (1000 km). Take a digital photograph of your sketch to including in your lab report.The deep linear features, the lowest points on Earth, are called ocean trenches.Using Google Earth, "fly to" Challenger Deep, the deepest place on Earth (once Google Earth gets you there, you may have to zoom out to see where you are). Where is it?Challenger Deep reaches 11 km (11,000 meters, equivalent to 36,000 ft) below sea level. Which is greater, the elevation of Mt Everest above sea level (see Question 3), or the depth of Challenger Deep below sea level, and by how much?In the space below, give the locations of three other ocean trenches on Earth.C. Seismic PatternsAn earthquake is a vibration of Earth caused by the sudden release of energy, usually as an abrupt breaking of rock along planar fractures called faults.Earthquakes originate at a point called the focus (or hypocenter) which is not at the surface of the earth, but instead at some depth within the earth. The epicenter of an earthquake is the point directly above the focus on either the land surface or seafloor; the depth of an earthquake has nothing to do with water depth, but instead is the depth in the solid earth from epicenter to focus.Only rocks that are cold and brittle (the earth's lithosphere) can be broken in earthquakes. Rocks that are hot and ductile will stretch and deform slowly over time without breaking (the earth's asthenosphere) – and thus do not produce earthquakes. So observing where earthquakes occur, both horizontally and with depth, tells us something about where stress is concentrated, and also about the material properties of the earth. (Source: https://www.windows2universe.org/earth/geology/quake_1.html (Links to an external site.))Expand the Lab 1/Earthquakes item and click "on" the "Twenty years of large earthquakes" layer to show the epicenters of large earthquakes (those with magnitudes >= 6.0) during a 20-year period.Describe any patterns you see in the distribution of earthquake epicenters over the Earth's surface - do they form lines, arcs, circles or clusters? Are patterns connected or disconnected?Look closely at and around the Earth's ridges and trenches. The earthquake depth patterns associated with these features are different. Complete the chart below:In the vicinity of ridges.(Scan 1500km or so on either side)In the vicinity of trenches.(Scan ~1500 or so km on either side)Describe the depth or range of depths of earthquakes, and the distribution (symmetric or asymmetric?)Is there any pattern to the depth distribution?Using earthquake depths as evidence, is the Earth's lithosphere thicker in the vicinity of ridges or in the vicinity of trenches? Justify your answer.D. Volcano PatternsA volcano is an opening in the Earth's surface through which melted rock (magma), volcanic ash and/or gases escape from the interior of the Earth.Leaving the earthquake layer on, click on the Active Volcanoes layer. Describe the relationship between the locations of most active volcanoes and locations of earthquakes:E. Plate BoundariesThe theory of plate tectonics holds that the Earth's lithosphere is broken into a finite number of jigsaw puzzle-like pieces, or plates, which more relative to one another over a plastically-deforming (but still solid) asthenosphere. The boundaries between plates are marked by active tectonic features such as earthquakes, volcanoes, and mountain ranges and there is (relatively) little tectonic activity in the middle of plates. Unclick all the layers, and then click on the "plate boundary model" layer (click the box to show it and then click the + or arrow to expand the legend). This shows plate boundaries and the names of major plates.Find the boundary between the African and South American platesWhere is this plate boundary, relative to the coastlines of Africa and South America?Now click the other layers on and off so that you can see relationships between plate boundaries and these features. If you did not have the "plate boundary layer" available to you, how could you determine where this plate boundary was? Be sure to consider topography/bathymetry as well as the earthquake and volcano layers. List several ways and be specific.Travel westward across the South American plate to its boundary with the Nazca plateWhere is this plate boundary, relative to South America?If you did not have the "plate boundary layer" available to you, how could you determine where this plate boundary was? List several ways and be specific.F. Plate motionMotion across the mid-Atlantic ridge: the South American plate vs. the African plateTurn on the "Seafloor age" and the "Plate Boundary" Google Earth (GE) layers. The "Seafloor age" layer shows the ages of volcanic rocks that have erupted and cooled to form the ocean floor. Focus on the Atlantic Ocean. Note that the age bands generally run parallel to the spreading ridges. Seafloor age is a critical piece of evidence for plate tectonics; these are used to reconstruct how ocean basins have developed over time and predict how they may evolve in the future.How many million years (abbreviated Ma) does each colored band represent?On average, continental crust is 2 billion years old; the oldest rocks are 3.8 billion years old, and some of the grains in those rocks are even older. What is the age of the oldest seafloor? _______________________________On average, which is oldest – the continents or the ocean basins? _________________Find the South American plate, the African plate, and the Mid-Atlantic Ridge that marks the boundary between them. What happens to the age of the seafloor as distance increases away from the Mid-Atlantic Ridge?Is crust being created or destroyed at this plate boundary (and other spreading ridges)?Is this plate boundary divergent, convergent, or transform? ________________Focus on the northern Atlantic Ocean, near the east coast of the US and the northwest coast of Africa. How long ago did the northern Atlantic Ocean begin to open up or start spreading? Describe your reasoning.Did the northern Atlantic Ocean basin start to open at the same time as the southern Atlantic Ocean basin? How much older or younger is the northern Atlantic basin than the southern Atlantic basin? Describe your reasoning.G. Putting it all together:Prepare a report documenting this lab activity. Your report should discuss how plate tectonic theory relates to earthquakes, volcanoes, and the bathymetry (sea floor topography) of oceans. Along the way, include answers to all of the questions in this lab. 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