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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

Find ƒ'(x), ƒ"(x), and f(3)(x) for the following functions. f(x) = 5x4 + 10x3 + 3x + 6
Compute the derivative of the following functions. g(x) X e³x
Evaluate the derivative of the following functions at the given point.c = 2√s - 1; s = 25
Find y" for the following functions.y = cot x
Calculate the derivative of the following functions.sin(sin (ex))
Compute the derivative of the following functions.y = 3x2 - 2x + e-2x
Find ƒ'(x), ƒ"(x), and f(3)(x) for the following functions. f(x) = 3x2 + 5ex
Use the graph of f to sketch a graph of f'. УА 5 12 y=f(x) X
Determine the slope of the following curves at the given point.x2/3 + y2/3 = 2; (1, 1)
Evaluate the derivative of the following functions at the given point.A = πr2; r = 3
Find y" for the following functions.y = tan x
Calculate the derivative of the following functions.sin2(e3x+1)
Find the values of the following derivatives using the table.a.b.c. 1 f(x) 3 f'(x) 7 9 5 x 3 1 9 7 9 5 9 5 5 3 7 7 1 3 1 9 5 3 1 7
Calculate the derivative of the following functions.sin5(cos 3x)
Compute the derivative of the following functions.ƒ(x) = (1–2x)e-x
Determine the slope of the following curves at the given point.(x + y)2/3 = y; (4, 4)
Calculate the derivative of the following functions. √x + √x
Find y', y", and y"' for the following functions.y = √x + 2 (x - 3)
Find y" for the following functions.y = cos θ sin θ
Calculate the derivative of the following functions.cos4 (7x3)
Calculate the derivative of the following functions. √x + √x + √x
Calculate the derivative of the following functions.tan(e√3x)
Compute the derivative of the following functions.A = 2500e0.075t
Suppose ƒ(3) = 1 and ƒ'(3) = 4. Let g(x) = x2 + ƒ(x) and h(x) = 3ƒ(x).a. Find an equation of the line tangent to y = g(x) at x = 3.b. Find an equation of the line tangent to y = h(x) at x =
Determine the slope of the following curves at the given point.xy5/2 + x3/2y = 12; (4, 1)
Calculate the derivative of the following functions.(1 - e-0.05x)-1
Use the table to find the following derivatives. 1 f'(x) 3 g'(x) 2 X 2 5 4 3 2 3 درا 4 1 1 5 4 5
Find dy/dx for the following functions. y = sin x 1 + cos x
Use the table to find the following derivatives. 1 f'(x) 3 g'(x) 2 X 2 5 4 3 2 3 درا 4 1 1 5 4 5
Use the table to find the following derivatives. 1 f'(x) 3 g'(x) 2 X 2 5 4 3 2 3 درا 4 1 1 5 4 5
Find dy/dx for the following functions. y || 1 2 + sin x
Compute the derivative of the following functions. g(x) = (x + 1)e* x 2
Calculate the derivative of the following functions.ƒ(g(x2)), where f and g are differentiable for all real numbers
Compute the derivative of the following functions. h(x) = (x1)(2x² - 1) x³ - 1
Calculate the derivative of the following functions.[ƒ(g(xm))]n, where f and g are differentiable for all real numbers, and m and n are integers
Calculate the derivative of the following functions.y = 4 log3 (x2 - 1)
Find dy/dx for the following functions. 2 cos x 1 + sin x
Calculate the derivative of the following functions.y = log10x
Compute the derivative of the following functions. h(x) xex x + 1
Find dy/dx for the following functions.y = x cos x sin x
Calculate the derivative of the following functions.y = cos x ln (cos2 x)
Find dy/dx for the following functions. y X COS X 1 + x³
Find dy/dx for the following functions. y 1- COS X 1 + cos x
Find ƒ'(x), ƒ''(x), and ƒ''(x). f(x) X
Compute the derivative of the following functions. h(x) = (x + 1) x² ex
The height in feet of a rocket above the ground is given bya. Graph the height function and describe the motion of the rocket.b. Find the velocity of the rocket.c. Graph the velocity function and
a. Find the equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line. y 1 + 2 sin x; x TT 6
Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.ƒ(x) = 12x - 16
Use a calculator to approximate the following limits. e³x lim 0-x - 1 X
a. Find the equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line. y csc x; x = = TT 4
Calculate the derivative of the following functions.y = 1/log4 x
Use a calculator to approximate the following limits. n ( ₁ + ²)² lim 1 +
a. Find the equation of the line tangent to the following curves at the given value of x.b. Use a graphing utility to plot the curve and the tangent line. y COS X 1 - cos x X = ㅠ 3
Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.ƒ(x) = x-1/3
Use a calculator to approximate the following limits. lim x x +0x
Calculate the derivative of the following functions.y = log2 (log2 x)
Find the following higher-order derivatives. d² 2 dx² (log10 x)
Use a calculator to approximate the following limits. (-)* lim x0+x
Find the following higher-order derivatives. x=1 (2+ X) EXP d³
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Find ƒ'(x), ƒ''(x), and ƒ''(x).ƒ(x) = x2 e3x
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Find the following higher-order derivatives. d³ dx3 (x² In x)
Find the following higher-order derivatives. up dxn (2x)
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Suppose f(2) = 2 and ƒ(2) = 3. Let a. Find an equation of the line tangent to y = g(x) at x = 2.b. Find an equation of the line tangent to y = h(x) at x = 2. g(x) = x² f(x) and h(x) = f(x) x - 3'
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Use the definition of the derivative and the trigonometric identity cos (x + h) = cos x cos h - sin x sin h d to prove that - (cos x) = -sin x. dx
Use the following table to find the given derivatives. X f(x) 1 5 الا 3 4 2 زرا 2 3 4 3 5 2 4 2 5 ن 3 نیا 4 5 2 1 1 4 3 1 1 5 الا
Starting with a full tank of gas, the distance traveled by a particular car is D(g) = 0.05g2 + 35g, where D is measured in miles and g is the amount of gas consumed in gallons.a. Compute dD>dg.
Find d2y/dx2 for the following functions.y = e-2x2
a. Use the graph of y = sin x (see figure) to sketch the graph of the derivative of the sine function. b. Based upon your graph in part (a), what function appears to y 1 y = sin x 2T X
a. Use the definition of the derivative to show thatb. Show that the limit in part (a) is equal to 2.c. Use parts (a) and (b) to find the derivative of f(x) = e2x. d dx (²x) = ²x. lim e2. h→0 e2h
When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population iswhere t is
Calculate the derivative of the following functions (i) Using the fact that bx = ex ln b and (ii) By using logarithmic differentiation. y = (x2 + 1)x
Calculate the derivative of the following functions (i) Using the fact that bx = ex ln b and (ii) By using logarithmic differentiation.y = 3x
Use the definition of the derivative to evaluate the following limits. In x - 1 e lim xe X
Calculate the derivative of the following functions (i) Using the fact that bx = ex ln b and (ii) By using logarithmic differentiation.y = g(x)h(x)
The relative growth rate r of a function f measures the rate of change of the function compared to its value at a particular point. It is computed as r(t) = ƒ'(t)/ƒ(t).a. Confirm that the relative
Use the definition of the derivative to evaluate the following limits. lim h→0 In (eh) - 8 h
Use the definition of the derivative to evaluate the following limits. lim h→0 3+h (3 + h)³¹h - 27 h
Find dy/dx, where (x2 + y2)(x2 + y2 + x) = 8xy2.
Find dy/dx, where √3x7 + y2 = sin2 y + 100xy.
Find d2y / dx2, where √y + xy = 1.
The population of the world reached 6 billion in 1999 (t = 0). Assume Earth’s carrying capacity is 15 billion and the base growth rate is r0 = 0.025 per year.a. Write a logistic growth
The logistic model can be used for situations in which the initial population P0 is above the carrying capacity K. For example, consider a deer population of 1500 on an island where a large fire
Use the definition of the derivative to evaluate the following limits. 5 - 25 lim x 2 x 2 -
Find (f-1)'32 if ƒ (x) = x3 + x + 1.
Find y" for the following functions.y = sec x csc x
Use the graph of f to sketch a graph of f'. -2 УА 2 y = f(x) 2 X
Find y', y", and y"' for the following functions.y = sin √x
Determine the slope of the following curves at the given point.xy1/3 + y = 10; (1, 8)
Use the graph of ƒ in the figure to find the following values, if they exist.a. h(2)b.c. h(4)d.e. lim h(x) x-2
The position of an object moving along a line is given by the function s(t) = -4.9t2 + 30t + 20. Find the average velocity of the object over the following intervals.a. [0, 3]b. [0, 2]
Use the graph of ƒ in the figure to find the following values, if they exist.a. ƒ(2)b.c.d. lim f(x) x-2
Suppose Findwhere ƒ(x) ≤ g(x) ≤ h(x), for all x. lim f(x) = lim h(x) = 5. x→2 x-2
Compute the values ofin the following table and use them to discuss f(x) x + 1 (x - 1)² De
Suppose p and q are polynomials. If and q(0) = 2, find p(0). p(x) lim x-0 q (x) 10
If find lim f(x) = -8, x-2

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