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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = x4 + 4x3 + 4x2 + 1
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = 3x4 + 8x3 – 18x2 + 5
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.y = 6x – 9
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = x + 3/x – 4
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.y = √x2 + 1
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.y = x√9 – x2
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = x2/3
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = (x + 1)4/5
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = ln 5x2 + 4/x2 + 1
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = xe–3x
A manufacturer of CD players has determined that the profit P(x) (in thousands of dollars) is related to the quantity x of CD players produced (in hundreds) per month bya. At what production levels
For each function, find (a) The critical numbers; (b) The open intervals where the function is increasing; (c) The open intervals where it is decreasing.f(x) = xex2–3x
The annual unemployment rates of the U.S. civilian noninstitutional population for 1990–2009 are shown in the graph. When is the function increasing? Decreasing? Constant? Rate (%) 10 98765432 1990
The graph shows the amount of air pollution removed by trees in the Chicago urban region for each month of the year. From the graph we see, for example, that the ozone level starting in May increases
The percent of concentration of a drug in the bloodstream x hours after the drug is administered is given bya. On what time intervals is the concentration of the drug increasing?b. On what intervals
The figure shows estimated totals of nuclear weapons inventory for the United States and the Soviet Union (and its successor states) from 1945 to 2010.a. On what intervals were the total inventories
Use Equation (1) to prove that the electric field of the disk (Equation (3)) is obtained from the voltage of the disk (Equation (2)).Equations. E = AP dz' (1)
Find derivatives of the functions defined as follows.y = e4x
Find the derivative of each function defined as follows.y = 12x3 – 8x2 + 7x + 5
Apply Equation (1) to the voltage of a point charge (Equation (4)) to obtain the electric field of a point charge (Equation (5)).Equations E = dᏙ dz' (1)
Find derivatives of the functions defined as follows.y = e–2x
Determine whether each of the following statements is true or false, and explain why.The derivative of a sum is the sum of the derivatives.
Find the derivative of each function defined as follows.y = 8x3 – 5x2 – x/12
Find the derivative of each function defined as follows.y = 3x4 – 6x3 + x2/8 + 5
Show that the electric field in Equation (7) results from the electric potential in Equation (6).Equations. V = k₁ (R - z z+ z² 2R (6)
Find derivatives of the functions defined as follows.y = –8e3x
Determine whether each of the following statements is true or false, and explain why.The derivative of a product is the product of the derivatives.
Find the derivative of each function defined as follows.y = 5x4 + 9x3 + 12x2 – 7x
Find derivatives of the functions defined as follows.y = 1.2e5x
Sometimes for z very, very close to the disk, the third term in Equation (6) is so small that it can be dismissed. Show that the electric field is constant for this case. V = k₁ (R - z z+ 2R (6)
Determine whether each of the following statements is true or false, and explain why.The marginal cost function is the derivative of the cost function.
Find the derivative of each function defined as follows.f(x) = 6x3.5 – 10x0.5
Find derivatives of the functions defined as follows.y = –16e2x+1
Use a graphing calculator or Wolfram Alpha (which can be found at www.wolframalpha.com) to recreate the graphs of the functions in Figures 14 and 15.Figure 14Figure 15 Voltage
Determine whether each of the following statements is true or false, and explain why.The chain rule is used to take the derivative of a product of functions.
Find the derivative of each function defined as follows.f(x) = –2x1.5 + 12x0.5
Find derivatives of the functions defined as follows.y = –4e–0.3x
Use a spreadsheet to create a table of values for the functions displayed in Figures 14 and 15. Compare the three voltage functions and then compare the three electric field functions.Figure 14Figure
Find derivatives of the functions defined as follows.y = e–x2
Determine whether each of the following statements is true or false, and explain why.The derivative of ln|x| is the same as the derivative of ln x.
Find the derivative of each function defined as follows.y = –100√x – 11x2/3
Determine whether each of the following statements is true or false, and explain why.The only function that is its own derivative is ex.
Find the derivative of each function defined as follows.y = 8√x + 6x3/4
Find derivatives of the functions defined as follows.y = ex2
In Exercises 7–14, find f[g(x)] and g[f(x)].f(x) = x/8 + 7; g(x) = 6x – 1
Determine whether each of the following statements is true or false, and explain why.The derivative of 10x is x10x-1.
In Exercises 7–14, find f[g(x)] and g[f(x)].f(x) = –8x + 9; g(x) = x/5 + 4
Find the derivative of each function defined as follows.y = 10x–3 + 5x–4 – 8x
Find derivatives of the functions defined as follows.y = 3e2x2
In Exercises 7–14, find f[g(x)] and g[f(x)].f(x) = 1/x; g(x) = x2
Determine whether each of the following statements is true or false, and explain why.The derivative of ln kx is the same as the derivative of ln x.
Find the derivative of each function defined as follows.y = 5x‾5 – 6x–2 + 13x–1
Find derivatives of the functions defined as follows.y = –5e4x3
In Exercises 7–14, find f[g(x)] and g[f(x)].f(x) = 2/x4; g(x) = 2 – x
Determine whether each of the following statements is true or false, and explain why.The derivative of log x is the same as the derivative of ln x.
In Exercises 7–14, find f[g(x)] and g[f(x)]. f(x)=√x + 2; g(x) = 8x² - 6
Find the derivative of each function defined as follows.f(t) = 7/t – 5/t3
Find derivatives of the functions defined as follows.y = 4e2x2–4
Find the derivative of each function defined as follows. f(t) 14 = + t 12 + V2
In Exercises 7–14, find f[g(x)] and g[f(x)]. f(x) = 9x² - 11x; g(x) = 2√x + 2
Find derivatives of the functions defined as follows.y = –3e3x2+5
Find the derivative of each function defined as follows. y 6 7 + 3 = + √5 X
In Exercises 7–14, find f[g(x)] and g[f(x)]. f(x) = Vx + 1; g(x) = -1 X
Find derivatives of the functions defined as follows.y = xex
Find the derivative of each function defined as follows. -13 +
In Exercises 7–14, find f[g(x)] and g[f(x)]. 8 f(x) = — ; g(x) = √3-x X
Find derivatives of the functions defined as follows.y = x2e–2x
Find the derivative of each function defined as follows.p(x) = –10x–1/2 + 8x–3/2
Find derivatives of the functions defined as follows.y = (x + 3)2e4x
Find the derivative of each function defined as follows.h(x) = x–1/2 – 14x–3/2
Find derivatives of the functions defined as follows.y = (3x3 – 4x)e–5x
Find the derivative of each function defined as follows. y = 9 Vx
Find derivatives of the functions defined as follows.y = x2/ex
Find the derivative of each function defined as follows. y = -2 Vx
Find derivatives of the functions defined as follows.y = ex/2x + 1
Prove that if y = y0ekt where y0 and k are constants, then dy/dt = ky. (This says that for exponential growth and decay, the rate of change of the population is proportional to the size of the
Find the derivative of each function defined as follows. f(x) = x³ + 5 X
Find the derivative of each function defined as follows. g(x): = x² - 4x Vx
Find derivatives of the functions defined as follows.y = ex + e–x/x
Find derivatives of the functions defined as follows.y = ex – e–x/x
Find the derivative of each function defined as follows.g(x) = (8x2 – 4x)2
Find derivatives of the functions defined as follows.p = 10000/(9 + 4e–0.2t)
Find the derivative of each function defined as follows.y = (8x4 – 5x2 + 1)4
Find the derivative of each function defined as follows.h(x) = (x2 – 1)3
Find derivatives of the functions defined as follows.p = 500/12 + 5e–0.5t
Find the derivative of each function defined as follows.y = (2x3 + 9x)5
Find derivatives of the functions defined as follows.f(z) = (2z + e–z2)2
Find the derivative of each function defined as follows.k(x) = –2(12x2 + 5)–6
Find derivatives of the functions defined as follows.f(t) = (et2 + 5t)3
Find the derivative of each function defined as follows.f(x) = –7(3x4 + 2)–4
Find derivatives of the functions defined as follows.y = 73x+ 1
Find the derivative of each function defined as follows.s(t) = 45(3t3 – 8)3/2
Find derivatives of the functions defined as follows.y = 4–5x+2
Find the derivative of each function defined as follows.s(t) = 12(2t4 + 5)3/2
Find derivatives of the functions defined as follows.y = 3 · 4x2 + 2
Find each derivative. D 9x-1/2 + 2
Find the derivative of each function defined as follows.g(t) = –3√7t3 – 1
Find each derivative. 8 Dx D. [√³/₁ 3 X Vx³
Find derivatives of the functions defined as follows.y = –103x2– 4
Find the derivative of each function defined as follows.f(t) = 8√4t2 + 5)3/2

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