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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

An article in a medical journal says that a sunscreen with a sun protection factor (SPF) of 2 provides 50% protection against ultraviolet B (UVB) radiation, an SPF of 4 provides 75% protection, and
Find the average rate of change for each function over the given interval.y = -4x2 - 6 between x = 2 and x = 6
Determine whether each statement is true or false, and explain why.Limits at infinity or negative infinity, if they exist, correspond to horizontal asymptotes of the graph of the function.
Estimate the slope of the tangent line to each curve at the given point (x, y). (-2,2) y 4 -2 0 -2 +₁. X
Find the average rate of change for each function over the given interval.y = -3x3 + 2x2 - 4x + 1 between x = -2 and x = 1
Choose the best answer for each limit.(a) Is 5. (b) Is 6.(c) Does not exist. (d) Is infinite. If lim f(x) = 5 and lim f(x) = 6, then lim f(x) x-2 x-2+ x-2
Find the average rate of change for each function over the given interval.y = 2x3 - 4x2 + 6x between x = -1 and x = 4
In Exercises, choose the best answer for each limit.(a) Does not exist. (b) Is 6.(c) Is -∞. (d) Is ∞. If lim f(x)= x-4 lim f(x) x-4 = lim_ f(x) = 6, but f(4) does not exist, then x-4+
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = -2x + 5
Suppose the position of an object moving in a straight line is given by s (t) = t2 + 5t + 2. Find the instantaneous velocity at each time.t = 6
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = -4x2 + 9x + 2
Find the instantaneous rate of change for each function at the given value.ƒ(x) = x2 + 2x at x = 0
Let ƒ(x) = 5x2 - 3 and g(x) = -x2 + 4x + 1. Find the following.(a) ƒ(-2) (b) g(3) (c) ƒ(-k)(d) g(3m) (e) ƒ(x + h) (f) g(x + h)(g)(h) f(x +h)-f(x) h
Using the graph of f (x) in Figure 10, show the graph of a f (x) where a satisfies the given condition.1 < a Figure 10 y=f(x) X
For each function, find (a) f (4), (b) f (-1/2), (c) ƒ (a), (d) f (2/m), and (e) Any values of x such that f (x) = 1. f(x) = x-4 2x + 1 10 if x # if x || 112 112
The power of personal computers has increased dramatically as a result of the ability to place an increasing number of transistors onto a single processor chip. The following table lists the number
In 1960 in an article in Science magazine, H. Van Forester, P. M. Mora, and W. Amiot predicted that world population would be infinite in the year 2026. Their projection was based on the rational
The table gives the average monthly atmospheric concentration of carbon dioxide, a major contributor to climate change, in parts per million for December of selected years, as recorded at the Mauna
Biologists have long noticed a relationship between the area of a piece of land and the number of species found there. The following data shows a sample of the British Isles and how many vascular
A large cloud of radioactive debris from a nuclear explosion has floated over the Pacific Northwest, contaminating much of the hay supply. Consequently, farmers in the area are concerned that the
One formula for estimating the mass (in kg) of a polar bear is given by m(g) = e0.02 +0.062g-0.000165g2, where g is the axillary girth in centimeters. It seems reasonable that as girth
The concentration of a certain drug in the bloodstream at time t (in minutes) is given byc(t) = e-t - e-2t.Use a graphing calculator to find the maximum concentration and the time when it occurs.
The fact that Sifan Hassan ran the mile in 4 minutes 12.33 seconds in 2019 could be used to calculate her average speed (rate of change).Determine whether each statement is true or false, and explain
The derivative of ƒ(x) represents the instantaneous rate of change of y = ƒ(x) with respect to x.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.The limit of ƒ(x) as x → a describes what happens to ƒ(x) when x is near, but not equal to, the value a.
The SpaceX Falcon 9 rocket’s velocity exactly 1 minute after its launch from the Kennedy Space Center in 2020 is an example of an instantaneous rate of change.Determine whether each statement is
Determine whether each statement is true or false, and explain why.If a function exists at x = a, then the function is continuous atx = a.
The slope of the tangent line to ƒ(x) at x = a is equal to ƒ′(a).Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.For the limit of ƒ(x) as x → a to exist, ƒ must be defined at a.
Estimate the slope of the tangent line to each curve at the given point (x, y). 6 4. 2 0 (5, 3) 2. 4 6 X
Determine whether each statement is true or false, and explain why.If ƒ(x) approaches infinity as x →a, then exists. lim f(x) x→a
Determine whether each statement is true or false, and explain why.The average rate of change between two points is equal to the slope of the line segment connecting the two points.
Determine whether each statement is true or false, and explain why.An exponential function is continuous everywhere.
A derivative can exist at a point where the function does not exist.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.For the limit of ƒ(x) as x →a to exist, the limit from the left and the limit from the right must both exist and be the same.
Determine whether each statement is true or false, and explain why.Velocity is always positive.
Determine whether each statement is true or false, and explain why.A logarithmic function is continuous everywhere.
Estimate the slope of the tangent line to each curve at the given point (x, y). y 2₁- 2+₁ (2, 2) 2 A+++ X
Find the average rate of change for each function over the given interval.y = x2 + 2x between x = 1 and x = 3
If ƒ(x) is continuous at a point, then its derivative ƒ′(x) always exists at that point.Determine whether each statement is true or false, and explain why.
Determine whether each statement is true or false, and explain why.If the limit has the indeterminate form (0/0), then the limit does not exist.
In Exercises, choose the best answer for each limit.(a) Is -1. (b) Does not exist.(c) Is infinite. (d) Is 1. If lim f(x) = lim f(x) = -1, but f(2)= 1, then lim f(x) x-2 x->2t x-2
Estimate the slope of the tangent line to each curve at the given point (x, y). -4... -2. (-3,-3) 0 -2 -2 2 x
Estimate the slope of the tangent line to each curve at the given point (x, y). 0 $ -2 (3,-1). LI
Estimate the slope of the tangent line to each curve at the given point (x, y). y 6 4 12 0 2. IT (4,2) 4 6 L X
In Exercises, for all values of x = a where the function is discontinuous, determine if the discontinuity is removable or nonremovable.Exercise 5In Exercises, find all values x = a where the function
Find the average rate of change for each function over the given interval.y = √x between x = 1 and x = 4
In Exercises, for all values of x = a where the function is discontinuous, determine if the discontinuity is removable or nonremovable.Exercise 6In Exercises, find all values x = a where the function
Find the average rate of change for each function over the given interval.y = √3x - 2 between x = 1 and x = 2
In Exercises, for all values of x = a where the function is discontinuous, determine if the discontinuity is removable or nonremovable.Exercise 10In Exercises, find all values x = a where the
Find the average rate of change for each function over the given interval.y = ex between x = -2 and x = 0
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = 3x - 7
Find the average rate of change for each function over the given interval.y = ln x between x = 2 and x = 4
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist.
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. f(x)
Suppose the position of an object moving in a straight line is given by s (t) = t2 + 5t + 2. Find the instantaneous velocity at each time.t = 1
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. f(x)
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.f (x) = 6x2 - 5x - 1
Suppose the position of an object moving in a straight line is given by s(t) = 5t2 - 2t - 7. Find the instantaneous velocity at each time.t = 2
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = 12/x
Suppose the position of an object moving in a straight line is given by s(t) = 5t2 - 2t - 7. Find the instantaneous velocity at each time.t = 3
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist.
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = 3/x
Decide whether each limit exists. If a limit exists, find its value. lim g(x) X→ ∞ -2 g(x). 6 3 -3 0 4 X
Suppose the position of an object moving in a straight line is given by s (t) = t3 + 2t + 9. Find the instantaneous velocity at each time.t = 1
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = √x
Suppose the position of an object moving in a straight line is given by s (t) = t3 + 2t + 9. Find the instantaneous velocity at each time.t = 4
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = -3 √x
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist.p(x) =
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = 2x3 + 5
Find the instantaneous rate of change for each function at the given value.s(t) = -4t2 - 6 at t = 2
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist.q(x) =
Use the table of values to estimate lim f(x). x-1 X²
Using the definition of the derivative, find f′(x). Then find f′(-2), f′(0), and f′(3) when the derivative exists.ƒ(x) = 4x3 - 3
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. r(x) 5
In Exercises, complete the tables and use the results to find the indicated limits. 2x² - 4x + 7, find lim f(x). x-1 If f(x) = 2x²
Find the instantaneous rate of change for each function at the given value.g(t) = 1 - t2 at t = -1
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) = x2 +
Find the instantaneous rate of change for each function at the given value.F(x) = x2 + 2 at x = 0
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) = 6 -
In Exercise, why doeseven though ƒ(1) = 2? lim f(x) = 1, x→1
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. k(x) =
In Exercises, complete the tables and use the results to find the indicated limits. If k(x) = x³ - 2x - 4 x = 2 - find lim k(x). x-2
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for each function at the given value.ƒ(x)
In Exercises, complete the tables and use the results to find the indicated limits. If f(x) = 2x³ + 3x² - 4x - 5 x + 1 -, find lim f(x). X→-1
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. j(x) =
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist. r(x) =
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) = 5/x;
In Exercises, complete the tables and use the results to find the indicated limits. If h(x) = √x - 2 x = 1 find lim h(x). x-1
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for each function at the given value.ƒ(x)
In Exercises, find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn’t exist.j(x) =
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) =
In Exercises, complete the tables and use the results to find the indicated limits. If f(x) = √x - 3 x - 3 find lim f(x). x-3
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for each function at the given value.ƒ(x)
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) =
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for each function at the given value.ƒ(x)
In Exercises, determine the interval over which the function is continuous. f(x) 3x - 2 x + 4
In Exercises, determine the interval over which the function is continuous. f(x) = eVI-x
For each function, find (a) The equation of the secant line through the points where x has the given values, and (b) The equation of the tangent line when x has the first value.ƒ(x) = √x
Explain the difference between the average rate of change of y as x changes from a to b, and the instantaneous rate of change of y at x = a.

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