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mathematics
precalculus 1st

Precalculus 1st Edition Jay Abramson - Solutions

For the following exercises, consider the function whose graph appears in Figure 3.Find an equation of the tangent line to the graph of f the indicated point: f(x) = 3x2 − 2x − 6, x = − 2 תי 43 -5-4-3-2-1 # 14 y 2345 III 3 4 5 IIIII Figure 3 # II X
For the following exercises, determine where the given function f(x) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities. f(x)= = x+2 x³ + 8
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = ln x2
For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiableot differentiable. y NH H
For the following exercise, use the given information to evaluate the limits: lim f(x) = 3, lim g(x) = 5
For the following exercises, consider the graph of the function f and determine where the function is continuous/ discontinuous and differentiable/not differentiable. x
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x)= = x² - 1 x² 3x + 2 ; a = 1
For the following exercise, use the given information to evaluate the limits: lim f(x) = 3, lim g(x) = 5
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = e2x
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x)= 1-x² x²-3x+2 ; a = 1
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x) = 10 10x² x² 3x + 2 - - ; a = 1
For the following exercises, evaluate the following limits. lim cos(x) x-2
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f(−1) D -6-5-4-3-2-1 y 4 2- -7 -8- a V 40+ Figure 20 I f(x) 2 3 4 5 6 x
For the following exercises, find the derivative of each of the functions using the definition: f(x) = 2x − 8 lim h→0 f(x+h)-f(x) h
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x)= X 6x²5x - 6 ; a 3 MIN 2
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = sec(x) − 3.
For the following exercises, evaluate the following limits. lim sin(x) x → 2
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.Determine the values of b and c such that the following
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f(0) -6-5-4-3-2-1 III y -4- 3 -6- -7+ 8- 10+ 2 3 56 V f(x) Figure 20 -X
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f(1) -6-5-4-3-2-1 N -2 4 y -6- -7 -8 40 [D] If(xx) III. IIIII Figure 20 2 3 4 5 6 X
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = x2 + sin(x)
For the following exercises, find the derivative of each of the functions using the definition: f(x) = 4x2 − 7 lim h→0 f(x+h)-f(x) h
For the following exercises, evaluate the following limits. lim sin x 2 TT X
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f(2) 6-5-4-3-2- -5- 4- 2- 7 8- 1₂ I 40- Figure 20 If(x) 2 3 4 5 6 X
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x) = X 4x² + 4x+1¹a= | 1 2
For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a, determine which of the three conditions of continuity are satisfied at x = a and which are not. x = −3 y An Figure 15 x
For the following exercises, find the derivative of each of the functions using the definition: lim h→0 f(x+h)-f(x) h
For the following exercises, evaluate the following limits. f(x)= ) = [2x²+2x+1, x - 3, x ≤0 ; lim f(x) x>0²x=40+²
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f(3) -4- 3- -6-5-4-3-2-1 y ننا -8- f(x) 1 2 3 4 5 6 Figure 20 X
For the following exercises, use numerical evidence to determine whether the limit exists at x = a. If not, describe the behavior of the graph of the function near x = a. Round answers to two decimal places. f(x) = 2 x-4 ; a = ;a=4
For the following exercises, find the derivative of the function. f(x) = 4x − 6
For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a, determine which of the three conditions of continuity are satisfied at x = a and which are not. x = 2 y LA Figure 15
For the following exercises, evaluate the following limits. 2x²+2x+1, (x - 3, f(x)=₁ x≤0 x>0 x 0 ; lim_ f(x)
For the following exercises, find the derivative of each of the functions using the definition: lim h→0 f(x+h)-f(x) h
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f′(−1) Figure 20 L LLLL ∞ ∞ A HI A N 67. X y
For the following exercises, find the derivative of the function. f(x) = 5x2 − 3x
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value. 7tanx lim x-0 3x
For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of a, determine which of the three conditions of continuity are satisfied at x = a and which are not.x = 4 y TM.. Figure 15 x
For the following exercises, find the derivative of each of the functions using the definition: lim h→0 f(x+h)-f(x) h
For the following exercises, evaluate the following limits. f(x) = 2x² + 2x + 1, x < 0 (x - 3, ; lim f(x) x>0 x 0
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f′(0) -5- 3 -2 -6-5-4-3-2-1 2 5- -6- -8- -9- 10- 2 3 4 5 6 V f(x) Figure 20 X
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value. x² lim x 4 x 4
For the following exercises, find the derivative of each of the functions using the definition: f(x) = −x3 + 1 lim h→0 f(x+h)-f(x) h
For the following exercises, evaluate the following limits. lim x 4 Vx+5-3 x-4
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f′(1) 6-5-4-3-2-1 4 3- -2₂ Es ∞0 -8- II 1 2 3 4 5 6 V [ f(x) Figure 20 X
For the following exercises, use a graphing utility to graph the function f(x) = sin (12π/x) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.Which conditions for continuity fail at the point of discontinuity? -10 -in 5 Figure
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value. lim x →0 2sinx 4tanx
For the following exercises, use a graphing utility to graph the function f(x) = sin (12π/x) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.Evaluate f(0). -10 5 y Figure 16 5 10 x
For the following exercises, find the derivative of the function.Find the equation of the tangent line to the graph of f(x) at the indicated x value. f(x) = −x3 + 4x; x = 2.
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f′(2) -6-5-4-3-2-1 y 3- 21 45 -7- C f(x) 2 3 4 5 6 V DIXID Figure 20 X
For the following exercises, find the derivative of each of the functions using the definition: f(x) = x2 + x3 lim h→0 f(x+h)-f(x) h
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. lim x 0 eet
For the following exercises, evaluate the following limits. lim (2x-[x]) x → 2+
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. f(x) = x/|x|
For the following exercises, find the derivative of each of the functions using the definition: lim h0 f(x+h)-f(x) h
For the following exercises, evaluate the following limits. Vx+7-3 lim x 2 x²-x-2
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.f′(3) y -3- 6-5-4-3-2-1 5 -6 [I] f(x) -8 1 2 3 4 5 6 I V H III II 1 Figure 20 X
For the following exercises, use a graphing utility to graph the function f(x) = sin (12π/x) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.Solve for x if f(x) = 0. -10 -in 5 y Figure 16 -in 5 10
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. lim x → 0 ee x²
For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable. Given that the volume of a right circular cone is V = 1/3 πr2h and that a given cone has a
For the following exercises, use a graphing utility to graph the function f(x) = sin (12π/x) as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.What is the domain of f(x)? -10 +5 Figure 16 5 10 ➤X
For the following exercises, consider the function shown in Figure 17.At what x-coordinates is the function discontinuous? SETE in Figure 17 # 3 4 5
For the following exercises, evaluate the following limits. x lim x 3 x-9
For the following exercises, use Figure 20 to estimate either the function at a given value of x or the derivative at a given value of x, as indicated.Sketch the function based on the information below: f′(x) = 2x, f(2) = 4 y CRITE -6-5-4-3-2-1 -5 -6 A f(x) 40+ Figure 20 2 3 4 5 6 VE X
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. |x| lim x-0 X
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. x+1 lim x-1 x+1
For the following exercises, consider the function shown in Figure 17.What condition of continuity is violated at these points? -445 2. 01 03 3 P 10 y Figure 17 x
Consider the function shown in Figure 18. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated? Figure 18
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. x-5 lim x-5 5-x
For the following exercises, explain the notation in words. The volume f(t) of a tank of gasoline, in gallons, t minutes after noon. f(0) = 600
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. 1 lim x-1(x + 1)²
For the following exercises, explain the notation in words. The volume f(t) of a tank of gasoline, in gallons, t minutes after noon. f′(30) = −20
Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x = −7 and x = 1.
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. 1 lim x →1 (x - 1)³
For the following exercises, explain the notation in words. The volume f(t) of a tank of gasoline, in gallons, t minutes after noon. f(30) = 0
For the following exercises, explain the notation in words. The volume f(t) of a tank of gasoline, in gallons, t minutes after noon.f′(200) = 30
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. lim x 0 5 2 1 - ex
The graph of f(x) = sin(2x)/x is shown in Figure 20. Is the function f(x) continuous at x = 0? Why or why not? s 2:5 -45-4-35-325-215-1-0,5 40:5+ N 051525 Figure 20 35 4 45
For the following exercises, explain the notation in words. The volume f(t) of a tank of gasoline, in gallons, t minutes after noon.f(240) = 500
For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t2 + 80t.s(2) = 96
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit. According to the
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as x approaches a. If the function has a limit as x approaches a, state it. If not, discuss why there is no limit.Allow the speed of light,
For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t2 + 80t.s′(2) = 16
For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t2 + 80t.s(3) = 96
For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t2 + 80t.s′(3) = −16
For the following exercises, explain the functions in words. The height, s, of a projectile after t seconds is given by s(t) = −16t2 + 80t.s(0) = 0, s(5) = 0.
For the following exercises, the volume V of a sphere with respect to its radius r is given by V = 4/3πr3.Find the average rate of change of V as r changes from 1 cm to 2 cm.
For the following exercises, the volume V of a sphere with respect to its radius r is given by V = 4/3πr3.Find the instantaneous rate of change of V when r = 3 cm.
For the following exercises, find the average rate of change The height of a projectile is given by s(t) = −64t2 + 192t Find the average rate of change of the height from t = 1 second to t = 1.5 seconds. f(x+h)-f(x) h
For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x.Find the average change of the revenue function as x changes from x = 10 to x = 20.
For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x.Find R′(10) and interpret.
For the following exercises, the revenue generated by selling x items is given by R(x) = 2x2 + 10x.Find R′(15) and interpret. Compare R′(15) to R′(10), and explain the difference.
For the following exercises, the cost of producing x cellphones is described by the function C(x) = x2 − 4x + 1000. Find the average rate of change in the total cost as x changes from x = 10 to x = 15.
For the following exercises, use the definition for the derivative at a point to find the derivative of the functions.f(x) = −x2 + 4x + 7 x = a, lim x→a f(x) - f(a) x - a
For the following exercises, the cost of producing x cellphones is described by the function C(x) = x2 − 4x + 1000.Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16th cellphone.
For the following exercises, use the definition for the derivative at a point to find the derivative of the functions.f(x) = 5x2 − x + 4 x = a, lim x→a f(x) - f(a) x - a
For the following exercises, the cost of producing x cellphones is described by the function C(x) = x2 − 4x + 1000.Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st cellphone.
For the following exercises, use the definition for the derivative at a point to find the derivative of the functions.f(x) = −4/3 − x2 x = a, lim x→a f(x) - f(a) x - a
For the following exercises, determine why the function f is discontinuous at a given point a on the graph. State which condition fails. _f(x) = X, 2x, x = 3 x=3 a=3
For the following exercises, determine why the function f is discontinuous at a given point a on the graph. State which condition fails. 0 = (5 f(x) 5, x=0 [3, x=0 a=0

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