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mathematics
calculus with applications

Questions and Answers of Calculus With Applications

Each graphing calculator window shows the graph of a function f(x) and its derivative function f′(x). Decide which is the graph of the function and which is the graph of the derivative.
Each graphing calculator window shows the graph of a function f(x) and its derivative function f′(x). Decide which is the graph of the function and which is the graph of the derivative.
Determine whether each of the following statements is true or false, and explain why.The derivative is a function.
Determine whether each of the following statements is true or false, and explain why.The slope of the tangent line gives the average rate of change.
Each graphing calculator window shows the graph of a function f(x) and its derivative function f′(x). Decide which is the graph of the function and which is the graph of the derivative.
Determine whether each of the following statements is true or false, and explain why.The derivative of a function exists wherever the function is continuous.
Sketch the graph of the derivative for each function shown. -6.. f(x) 4 -2. 2 A 12 6 X
Is a derivative always a limit? Is a limit always a derivative? Explain.
Is every continuous function differentiable? Is every differentiable function continuous? Explain.
Sketch the graph of the derivative for each function shown. -6. -4 -2 f(x) -2 2 4. 19 X
Give two applications of the derivative f'(x) = lim h→0 f(x +h)-f(x) h
Describe how to tell when a function is discontinuous at the real number x = a.
Sketch the graph of the derivative for each function shown. f(x) -6-4. -2. 2 ନ 2 4 6 X
Sketch the graph of the derivative for each function shown. -6-4. अ f(x) 1 2 4 6 X
Decide whether the limits in Exercises exist. If a limit exists, find its value.(a)(b)(c)(d) ƒ(-3) lim f(x) x-3
Sketch the graph of the derivative for each function shown. -6-4 f(x) + -2 -2 2; 4.6 X
Sketch the graph of the derivative for each function shown. -6 -2 f(x) + 2 ..2. 6 X
Decide whether the limits in Exercises exist. If a limit exists, find its value.(a)(b)(c)(d) g(-1) lim_g(x) x-1
Decide whether the limits in Exercises exist. If a limit exists, find its value.(a)(b)(c)(d) ƒ(4) lim f(x) x-4
Decide whether the limits in Exercises exist. If a limit exists, find its value.(a)(b)(c)(d) h(2) lim h(x) x-2
Sketch the graph of the derivative for each function shown. f(x) ↑ -6 -4 -2. 2 4 6 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.
Sketch the graph of the derivative for each function shown. f(x). hm 1 -14 -3 X
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim_ g(x) X→-∞ g(x) -2 -2 2 X
Decide whether the limits in Exercises exist. If a limit exists, find its value. 2x + 7 lim x 6 x + 3
Sketch the graph of the derivative for each function shown. f(x) + -6 -4 -2.. 2 2 4 6 X
Sketch the graph of the derivative for each function shown. 2 f(x) 2 0 the X
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim f(x) X→∞⁰ -4 f(x) 3- دیا -3- 4 X
Decide whether the limits in Exercises exist. If a limit exists, find its value. 2x lim x 3 x + 5 3
The graph at the top of the next column shows how the body mass index-for-age percentile for boys varies from the age of 2 to 20 years.(a) Sketch a graph of the rate of change of the 95th percentile
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim X-2 x² + 3x - 10 x - 2
Decide whether the limits in Exercises exist. If a limit exists, find its value. x² - 16 4 lim X-4 X -
The growth remaining in sitting height at consecutive skeletal age levels is indicated below for boys. Sketch a graph showing the rate of change of growth remaining for the indicated years. Use the
Decide whether the limits in Exercises exist. If a limit exists, find its value. √x - 3 lim X9 X9
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim X-4 2x² + 3x - 20 - x + 4
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim x-3 →3 3x² 2x - 21 x - 3
The graph at the top of the next column shows the typical weight (in kilograms) of an English boy for his first 18 years of life.(a) Sketch the graph of the rate of change of weight with respect to
Decide whether the limits in Exercises exist. If a limit exists, find its value. √x - 4 lim x 16 x 16 -
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim 1118 3 m 100 8 + 3 X 6 2 x²
The graph below shows the discharge of water (in cubic feet per second) from the Otter Creek in Middlebury, VT, during March 2019. Sketch a graph of the rate of change in the discharge with respect
Decide whether the limits in Exercises exist. If a limit exists, find its value. lim X→∞0 2x² + 5 − 1 5x2
Identify the x-values where f is discontinuous. f(x) pv 3 | 0 | 1X3 x4 X
Decide whether the limits in Exercises exist. If a limit exists, find its value. X x² + 6x + 8 lim .3 x x³ + 2x + 1
Decide whether the limits in Exercises exist. If a limit exists, find its value. 9 lim x-00x² 4 10 X 6
Identify the x-values where f is discontinuous. Xj f(x) क X2 H 0 X3 X4 X
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist. f(x) = x² - 9 x + 3
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist. f(x) = -5 + x 3x(3x + 1)
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist. f(x) = 7 - 3x (1-x)(3 + x)
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist. f(x) = 9 x-6 x + 5
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x
Based on the U.S. Census population projections for 2016 to 2060, the projected Asian population (in millions) can be modeled by the exponential function A(t) = 14.8311.01592t, where t = 0
For the function shown in the sketch, give the intervals or points on the x-axis where the rate of change of ƒ(x) with respect to x is(a) positive; (b) negative; (c) zero. a f(x) 0 b c X
In Exercises, tell which graph, (a) or (b), represents velocity and which represents distance from a starting point.(a)(b) y 40 30 20 10 0 + 2 4 8
In Exercises, (a) Graph the given function, (b) Find all values of x where the function is discontinuous, and (c) Find the limit from the left and from the right at any values of x
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist.ƒ(x) = x2 + 3x - 4
In Exercises, tell which graph, (a) or (b), represents velocity and which represents distance from a starting point.(a)(b) y 25 20 15 10 5 0 2 4 6
The body mass of yearling bighorn sheep on Ram Mountain in Alberta, Canada, can be estimated byM(t) = 27.5 + 0.3t - 0.001t2where M1t2 is measured in kilograms and t is days since May 25.(a) Find the
Find each limit (a) By investigating values of the function near the point where the limit is taken and (b) By using a graphing calculator to view the function near the point. lim x-1 x² +
Find all x-values where the function is discontinuous. For each such value, give f (a) and lim x→a f (x) or state that it does not exist.ƒ(x) = 2x2 - 5x - 3
Find the average rate of change for the following on the given interval. Then find the instantaneous rate of change at the first x-value. У -6 3x - 5 from x = 4 to x = 9
Find each limit (a) By investigating values of the function near the point where the limit is taken and (b) By using a graphing calculator to view the function near the point. lim x--2 x²
Find the average rate of change for the following on the given interval. Then find the instantaneous rate of change at the first x-value. y x + 4 x - 1 from x = 2 to x = 5
Tell at what values of x the function ƒ(x) in Figure 9 from the previous section is discontinuous. Explain why it is discontinuous at each of these values.
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. f(x): = 1 -
Find the average rate of change for the following on the given interval. Then find the instantaneous rate of change at the first x-value.y = 6x3 + 2 from x = 1 to x = 4
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. f(x)
Find the average rate of change for the following on the given interval. Then find the instantaneous rate of change at the first x-value.y = -2x3 - 3x2 + 8 from x = -2 to x = 6
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. f(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) = 3x2
Let ƒ and g be differentiable functions such thatwhere c Z d. Determine(Choose one of the following.) Source: Society of Actuaries.(a) 0(b)(c) ƒ′(0) - g′(0)(d) c - d(e) c + d lim f(x) = c lim
Sketch the graph of the derivative for each function shown. + -6 -4 -2 f(x) 2 จ 2 4 6 x
Sketch the graph of the derivative for each function shown. -6 --4 -2 f(x) 2 -2 + +11 2 4 6 X
The graph shows the total cost C(x) to produce x tons of cement. (Recall that average cost is given by total cost divided by the number produced, or C(x) = C(x)/x.)(a) Find the value of x for which
Use the definition of the derivative to find the derivative of the following.ƒ(x) = 4x2 + 3x - 2
Use the definition of the derivative to find the derivative of the following.ƒ(x) = 5x2 - 6x + 7
In Exercises, find the derivative of the function at the given point (a) By approximating the definition of the derivative with small values of h (b) By using a graphing calculator to zoom
The graph below shows the projected number of people aged 65 and over in the United States with Alzheimer’s disease. Estimate and interpret the derivative in each of the following years.(a)
In Exercises, find the derivative of the function at the given point (a) By approximating the definition of the derivative with small values of h and (b) By using a graphing calculator to
A simplified income tax considered in the U.S. Senate in 1986 had two tax brackets. Married couples earning $29,300 or less would pay 15% of their income in taxes. Those earning more than $29,300
The unemployment rates in the United States for the years 1994–2014 are shown in the graph below. Sketch a graph showing the rate of change in the annual unemployment rates for this period. Use the
The growth remaining in sitting height for girls at consecutive skeletal age levels is indicated on the graph. Sketch a graph showing the rate of change of growth remaining for the indicated years.
Waverly Products has found that its revenue is related to advertising expenditures by the functionR(x) = 5000 + 16x - 3x2, where R(x) is the revenue in dollars when x hundred dollars are spent on
Suppose a gram of ice is at a temperature of -100°C. The graph shows the temperature of the ice as increasing numbers of calories of heat are applied. It takes 80 calories to melt one gram of ice at
The following figure, already shown in Section 2.1, Properties of Functions, shows the depth of a sperm whale as a function of time, recorded by researchers at the Woods Hole Oceanographic
The following graph shows how the body mass index-for-age percentile for girls varies from the age of 2 to 20 years.(a) Sketch a graph of the rate of change of the 95th percentile as a function of
A company charges $1.50 per lb when a certain chemical is bought in lots of 125 lb or less, with a price per pound of $1.35 if more than 125 lb are purchased. Let C(x) represent the cost of x lb.
In Chapter 1, we saw that the cost to fly x miles on American Airlines could be approximated by the equation C(x) = 0.00003964x + 191.16. Recall from the previous chapter that the average cost per
Use a graphing calculator to graph the function. (a) Determine the limit from the graph. (b) Explain how your answer could be determined from the expression for f (x). lim (1 + 5x¹/³ +
The spread of a virus is modeled byV(t) = -t2 + 6t - 4,where V(t) is the number of people (in hundreds) with the virus and t is the number of weeks since the first case was observed.(a) Graph
A company determines that t months after an employee starts working on the assembly line, the employee will be able to assemble Q(t) = 60 - 40e-0.3t products per day.(a) Determine the number of units
The cost (in dollars) for manufacturing a particular flashdrive isC(x) = 15,000 + 6x,where x is the number of flashdrives produced. Recall from the previous chapter that the average cost per
A company training program has determined that, on the average, a new employee produces P(s)Citems per day after s days of on-the-job training, whereFind and interpret P(s) = 63s S + 8
A company expects that sales of a new product can be approximated by the functionwhere S(t) is the number of units sold in month t .(a) Determine the number of units sold in month 1.(b) Determine the
In business finance, an annuity is a series of equal payments received at equal intervals for a finite period of time. The present value of an n-period annuity takes the formwhere R is the amount of
For some annuities encountered in business finance, called growing annuities, the amount of the periodic payment is not constant but grows at a constant periodic rate. Leases with escalation clauses
Researchers have developed a mathematical model that can be used to estimate the number of teeth N(t) at time t (days of incubation) for Alligator mississippiensis, where(a) Find N(65), the number of
To develop strategies to manage water quality in polluted lakes, biologists must determine the depths of sediments and the rate of sedimentation. It has been determined that the depth of sediment
The concentration of a drug in a patient’s bloodstream h hours after it was injected is given byFind and interpret A(h) = 0.17h h² + 2
Members of a legislature often must vote repeatedly on the same bill. As time goes on, members may change their votes. Suppose that p0 is the probability that an individual legislator favors an issue
The following table shows global estimated percentages in recent years of infants tested for HIV within two months of being born to women having HIV.(a) Plot the data on a graphing calculator,

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