In a study, a cancerous tumor was found to have a volume of
f (t) = 1.8253(1 - 1.6e-0.4196t)3
milliliters after t weeks, with t > 1. (Source: Growth, Development and Aging.)
(a) Sketch the graphs of f (t) and f ‘(t) for 1 ≤ t ≤ 15. What do you notice about the tumor’s volume?
(b) How large is the tumor after 5 weeks?
(c) When will the tumor have a volume of 5 milliliters?
(d) How fast is the tumor growing after 5 weeks?
(e) When is the tumor growing at the fastest rate?
(f) What is the fastest rate of growth of the tumor?
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> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
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> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (t) = t10 at t = 10 and t = 50
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> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = p2e-(p+3), p = 4
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = (77/p2) + 3, p = 1
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 400(116 - p2), p = 6
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 600e-0.2p, p = 10
> Consider an exponential decay function P(t) = P0e-λt, and let T denote its time constant. Show that, at t = T, the function P(t) decays to about one-third of its initial size. Conclude that the time constant is always larger than the half-life.
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> The wholesale price in dollars of one pound of pork is modeled by the function f (t) = 1.4 + .26t - .1t2 + .01t3, where t is measured in years from January 1, 2010. (a) Estimate the price in 2012 and find the percentage rate of increase of the price in 2
> The wholesale price in dollars of one pound of ground beef is modeled by the function f (t) = 3.08 + .57t - .1t2 + .01t3, where t is measured in years from January 1, 2010. (a) Estimate the price in 2011 and find the rate in dollars per year at which the
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> A function h (x) is defined in terms of a differentiable f (x). Find an expression for h(x). h(x) = xf (x)
> Differentiate the following functions. f (x) = -7ex/7
> Differentiate the following functions. f (x) = e√(x2+1)
> Differentiate the following functions. f (x) = e√x
> Differentiate the following functions. f (x) = e1/x
> The amount in grams of a certain radioactive material present after t years is given by the function P(t). Match each of the following answers with its corresponding question. Answers a. Solve P(t) = .5P(0) for t. b. Solve P(t) = .5 for t. c. P(.5) d. P
> The rate of growth of a certain cell culture is proportional to its size. In 10 hours a population of 1 million cells grew to 9 million. How large will the cell culture be after 15 hours?
> Differentiate the following functions. f (x) = eex
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> Find dy/dx if
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> Differentiate the following functions. f (x) = e(1+x)3
> The height of a certain plant, in inches, after t weeks is f (t) = 1/(.05 + e-0.4t). The graph of f (t) resembles the graph in Fig. 3. Calculate the rate of growth of the plant after 7 weeks. Figure 3: 30%/- 20 10 400 y=f"(1) 6.4 y=f(t) y = f'(0) 1
> Let T be the time constant of the curve y = Ce-λt as defined in Fig. 5. Show that T = 1>λ.
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> Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f (x) = (5x - 2)e1-2x
> Find d2y/dx2. y = x√x + 1
> Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f (x) = (4x – 1)/ex/2
> Many scientists believe there have been four ice ages in the past 1 million years. Before the technique of carbon dating was known, geologists erroneously believed that the retreat of the Fourth Ice Age began about 25,000 years ago. In 1950, logs from an
> Differentiate the following functions. f (x) = e4x2-x
> Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f (x) = (3 - 4x)/e2x
> Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f (x) = (1 - x)e2x
> Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f (x) = (1 + x)e-3x
> Simplify the function before differentiating. f (t) = √(e3x)
> Simplify the function before differentiating. f (x) = (ex + 5e2x)/ex
> Simplify the function before differentiating. f (t) = e3t(e2t - e4t)
> Simplify the function before differentiating. f (x) = 1/√ex
> Simplify the function before differentiating. f (x) = exe2xe3x
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> Simplify the function before differentiating. f (x) = (e3x)5
> Differentiate. f (x) = eex
> Differentiate the following functions. f (x) = e-3x-2
> Differentiate. f (x) = √(ex + 1)
> Differentiate. f (x) = (ex - e-x)/(ex + e-x)
> Differentiate. f (x) = (ex + e-x)/(ex - e-x)
> Differentiate. f(x) = එස
> Differentiate. f (x) = (x + 1/x)e2x
> Differentiate. f (t) = (t3 - 3t)e1+t
> Differentiate. f (t) = (t2 + 2et)et-1
> Differentiate the functions. y = x4 + 4 4√x
> Find d2y/dx2. y = (x2 + 1)4
> Differentiate. f (t) = 2et/2 - .4e0.001 t
> Differentiate. f (t) = 4e0.05t - 23e0.01t
> Differentiate the following functions. f (x) = 10e(-x – 2)/5
> Differentiate the following functions. f (x) = e2x+3
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> Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the
> Suppose that in Boston the wholesale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x + 8p = 328. If there are 4 thousand crates available today at a price of $25 per crate,