How many of the following characteristics can two different logarithmic functions share, while still remaining different?
Domain
Range
y-intercept
Intervals of increase or decrease
Decide how many key applicable characteristics there are and explain with examples.
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Find equations of the following.
2(x − 9)2 + (y − 6)2 + (z − 1)2 = 10, (10, 8, 3)
(a) the tangent plane
(b) the normal line
(x(t), y(t), z(t))=
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Consider the unit sphere x2 +y2 +z2 = 1 and the cone (z+√2)2 = x2 +y2. Show that these surfaces are tangent where they intersect, that is, for a point on the intersection, these surfaces have the same tangent plane
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Find the solution of the following nonhomogeneous linear system
y1′ = y2 + 1 ,
y2′ =−y1 + t.
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What information about a graph can be found from the second derivative? Explain
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| Poisoned water leaks from a crack in a storage tower at an increasing rate of 6t L/day and flows into a pond containing 108 L of water. (Note: t is time in days) This leaking water contains 4 g/L of poison. Water in the pond is well-mixed, and 6t L/day leaves the pond, keeping the volume of water in the pond at 108 L. If the pond is initially full, and contains no poison, how many days later does the pond contain 144 g of poison? |
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Use implicit differentiation to find ∂z/∂x and ∂z/∂y if xz = cos (y + z).
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1.The graph of (x-1)^2+(y-1)^2=4 is shown on the right. Use implicit differentiation to find the slopes of the tangent lines at the points on the graph where x=2.2 . Check your answers by visually estimating the slopes on the graph in the figure.
2.Use implicit differentiation to find y′. Then evaluate y′ at (2,1)
x^4-16y^3=ln y
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Analysis of daily output by a factory employee shows that, on average, the number of units per hour produced after t hours on the job can be modeled by y = 70t + 0.5t 2 − t 3 where 0 ≤ t ≤ 8.
(a) Thinking about the context of this problem, explain why t-values in this model might be restricted to values between t = 0 and t = 8.
(b) Find all the critical numbers for this function. Hint: You may want to use the quadratic formula or a quadratic program on your calculator as part of this calculation.
(c) Which critical number(s) that you found in part (b) make sense in this problem? Explain your reasoning in the context of this problem.
(d) For the critical number(s) you stated in part (c), explain, in the context of this problem, what this critical number tells you.
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Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f (x, y) = x2 + xy + y2 + 8y
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