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mathematics
calculus

Questions and Answers of Calculus

The tangent line to the graph of f(x) at x = 1 is shown. On the tangent line, P is the point of tangency and A is another point on the line.(a) Find the coordinates of the points P and A.(b) Use the
Find (a) The derivative, by using the definition. (b) The instantaneous rate of change of the function at any value and at the given value. (c) The slope of the tangent at the given value. 1. f(x) =
Approximate f'(a) in the following ways. (a) Use the numerical derivative feature of a graphing calculator. (b) Use f(a + h) - f(a) / h with h = 0.0001 (c) Graph the function on a graphing
Use the given tables to approximate f'(a) as accurately as you can.1.2.
At each point A and B draw an approximate tangent line and then use it to complete parts (a) and (b).(a) Is f '(x) greater at point A or at point B? Explain.(b) Estimate f '(x) at point B.1.2.
A point (a, b) on the graph of y = f (x) is given, and the equation of the line tangent to the graph of f (x) at (a, b) is given. In each case, find f'(a) and f(a).1. (4, - 11); 7x - 3y = 612. (-1,
If the instantaneous rate of change of f(x) at (2, - 4) is 5, write the equation of the line tangent to the graph of f (x) at x = 2.
If the instantaneous rate of change of g (x) at (- 1, - 2) is 1/2, write the equation of the line tangent to the graph of g(x) at x = - 1.
(a) Over what interval(s) (a) through (d) is the rate of change of f (x) positive?(b) Over what interval(s) (a) through (d) is the rate of change of f (x) negative?Because the derivative of a
Given the graph of y = f(x) in Figure 9.27, determine for which x-values A, B, C, D, or E the function is (a) Continuous. (b) Differentiable.
(a) Find the slope of the tangent to the graph of f(x) at any point, (b) Find the slope of the tangent at the given point, (c) Write the equation of the line tangent to the graph of f(x) at the given
Suppose total cost in dollars from the production of x printers is given by C(x) = 0.0001x3 + 0.005x2 + 28x + 3000 Find the average rate of change of total cost when production changes (a) From 100
If an object is thrown upward at 64 feet per second from a height of 20 feet, its height S after t seconds is given by S(x) = 20 + 64t - 16t2 What is the average velocity in the (a) First 2 seconds
If the demand for a product is given by D(p) = 1000 / √p - 1 What is the average rate of change of demand when p increases from (a) 1 to 25? (b) 25 to 100?
If the total revenue function for a blender is R(x) = 36x - 0.01x2 where x is the number of units sold, what is the average rate of change in revenue R(x) as x increases from 10 to 20 units?
Suppose the figure shows the total cost graph for a company. Arrange the average rates of change of total cost from A to B, B to C, and A to C from smallest to greatest, and explain your choice.
The figure shows the percent of the U.S. population that was foreign-born for selected years from 1910 and projected to 2020.(a) Use the figure to find the average rate of change in the percent of
The revenue function for a sound system is R(x) = 300x - x2 dollars where x denotes the number of units sold. (a) What is the function that gives marginal revenue? (b) What is the marginal revenue if
Suppose the total revenue function for a blender isR(x) = 36x - 0.01x2 dollarswhere x is the number of units sold.(a) What function gives the marginal revenue?(b) What is the marginal revenue when
The monthly output at the Olek Carpet Mill is Q(x) = 15,000 + 2x2 units, (40 ≤ x ≤ 60) where x is the number of workers employed at the mill. If there are currently 50 workers, find the
Suppose that the demand for x units of a product is x = 10,000 - 100p where p dollars is the price per unit. Then the consumer expenditure for the product is E(p) = px = p(10,000 - 100p) = 10,000p -
Suppose that the profit function for the monthly sales of a car by a dealership is P(x) = 500x - x2 - 100 where x is the number of cars sold. What is the instantaneous rate of change of profit
Given f(x) = 2x - x2, find the average rate of change of f (x) over each of the following pairs of intervals. (a) [2.9, 3] and [2.99, 3] (b) [3, 3.1] and [3, 3.01] (c) What do the calculations in
If the total revenue function and the total cost function for a toy are R(x) = 2x and C(x) = 100 + 0.2x2 + x what is the instantaneous rate of change of profit if 10 units are produced and sold?
The highest recorded temperature in the state of Alaska was 100 F and occurred on June 27, 1915, at Fort Yukon. The heat index is the apparent temperature of the air at a given temperature and
In learning theory, receptivity is defined as the ability of students to understand a complex concept. Receptivity is highest when the topic is introduced and tends to decrease as time passes in a
Suppose the graph shows a manufacturer's total revenue, in thousands of dollars, from the sale of x cellular telephones to dealers.(a) Is the marginal revenue greater at 300 cell phones or at 700?
The graph shows a model for the number of millions of Social Security beneficiaries projected to 2030. The model was developed with data from the Social Security Trustees Report.(a) Was the
Given f(x) = x2 + 3x + 7, find the average rate of change of f (x) over each of the following pairs of intervals. (a) [1.9, 2] and [1.99, 2] (b) [2, 2.1] and [2, 2.01] (c) What do the calculations in
In the Procedure/Example box in this section, we were given f(x) = 4x2 and found f(x) = 8x. Find (a) The instantaneous rate of change of f (x) at x = 4. (b) The slope of the tangent to the graph of y
In Example 6 in this section, we were given f(x) = 3x2 + 2x + 11 and found f(x) = 6x + 2. Find (a) The instantaneous rate of change of f(x) at x = 6. (b) The slope of the tangent to the graph of y =
Let f(x) = 3x2 - 2x. (a) Use the definition of derivative and the Procedure/Example box in this section to verify that f(x) = 6x - 2. (b) Find the instantaneous rate of change of f (x) at x = -
Find the derivatives of the functions. 1. y = 4 2. f(s) = 6 3. f (t) = t 4. s = t2
The indicated points, find(a) The slope of the tangent to the curve, and(b) The instantaneous rate of change of the function.1. y = 7x2 + 2x + 1, x = 22. C(x) = 3x2 - 5, (3, 22)
Find the derivative of each function. 1. y = x-5 + x-8 - 3 2. y = x-1- x-2 + 13
Write the equation of the tangent line to each curve at the indicated point. As a check, graph both the function and the tangent line.1. y = x3 - 5x2 + 7 at x = 12. y = x4 - 4x3- 2 at x = 2
Find the coordinates of points where the graph of f (x) has horizontal tangents. As a check, graph f (x) and see whether the points you found look as though they have horizontal tangents. 1. f(x) = -
Find each derivative at the given x-value (a) With the appropriate rule (b) With the numerical derivative feature of a graphing calculator. 1. y = 5 - 2 √x at x = 4 2. y = 1 + 3x2/3 at x = - 8
Complete the following. (a) Calculate the derivative of each function with the appropriate formula. (b) Check your result from part (a) by graphing your calculated derivative and the numerical
The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually
Do the following. (a) Find f'(x) (b) Graph both f (x) and f'(x) with a graphing utility. (c) Use the graph of f'(x) to identify x-values where f'(x) = 0, f'(x) > 0, and f'(x) < 0. (d) Use the graph
Suppose that a wholesaler expects that his monthly revenue, in dollars, for an electronic game will be R(x) = 100x - 0.1x2, 0 ≤ x ≤ 800 where x is the number of units sold. Find his marginal
The total revenue, in dollars, for a commodity is described by the function R = 300x - 0.02x2 (a) What is the marginal revenue when 40 units are sold? (b) Interpret your answer to part (a).
1. According to Kleiber's law the metabolic rate q of the vast majority of animals is related to the animal's mass M according to q = kM3/4 where k is a constant. This means that a cat, with mass
The demand for q units of a product depends on the price p (in dollars) according toFind and explain the meaning of the instantaneous rate of change of demand with respect to price when the price
Suppose that the demand for a product depends on the price p according towhere p is in dollars. Find and explain the meaning of the instantaneous rate of change of demand with respect to price
Suppose that the total cost function, in dollars, for the production of x units of a product is given byC(x) = 4000 + 55x + 0.1x2Then the average cost of producing x items is(a) Find the
Suppose that the total cost function, in dollars, for a certain commodity is given byC(x) = 40,500 + 190x + 0.2x2where x is the number of units produced.(a) Find the instantaneous rate of change of
Suppose that for a certain city the cost C, in dollars, of obtaining drinking water that contains p percent impurities (by volume) is given by C = 120,000 / p - 1200 (a) Find the rate of change of
Suppose that the cost C, in dollars, of processing the exhaust gases at an industrial site to ensure that only p percent of the particulate pollution escapes is given by C(p) = 8100(100 - p) / p (a)
One form of the formula that meteorologists use to calculate wind chill temperature (WC) is WC = 35.74 + 0.6215t - 35.75s0.16 + 0.4275t s0.16 where s is the wind speed in mph and t is the actual air
The table below gives the U.S. consumer price index (CPI) for selected years from 2012 and projected to 2050. With the reference year as 2012, a 2020 CPI = 120.56 means goods and services that cost
The following table gives the online sales, in billions of dollars, from 2000 and projected to 2017.(a) Model these data with a power function E(t), where t is the number of years past 1990. Report
The table gives the U.S. population to the nearest million for selected years from 1950 and projected to 2050.(a) Find a cubic function P(t) that models these data, where P is the U.S. population in
The table shows U.S. gross domestic product (GDP) in billions of dollars for selected years from 2000 to 2070 (actual and projected).(a) Model these data with a cubic function g = g(t), where g is in
Find the derivative and simplify.1. y = (5x + 3)(x2 - 2x)2. s = (t4 + 1) (t3 - 1)
Find the indicated derivatives and simplify.1.2. 3. 4.
At the indicated point for each function, find(a) The slope of the tangent line, and(b) The instantaneous rate of change of the function.1.2.
Write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line.1. y = (9x2 - 6x
Use the numerical derivative feature of a graphing calculator to find the derivative of each function at the given x-value.1.2. 3. 4.
Complete the following.(a) Find the derivative of each function, and check your work by graphing both your calculated derivative and the numerical derivative of the function.(b) Use your graph of the
(a) Find f'(x).(b) Graph both f (x) and f'(x) with a graphing utility.(c) Identify the x-values where f'(x) = 0, f '(x) > 0, and f'(x) < 0.(d) Identify x-values where f (x) has a maximum point
Prove the Quotient Rule for differentiation. Add [- u(x) ∙ v(x) + u(x) ∙ v(x)] to the expanded numerator and use steps similar to those used to prove the Product Rule.
Use the Quotient Rule to show that the Powers of x Rule applies to negative integer powers. That is, show that (d / dx)xn = nxn-1 when n = - k, k > 0, by finding the derivative of f (x) = 1 / (xk).
1. If the cost C (in dollars) of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by C(p) = 8100p / 100 - p find the rate of change of C with
Suppose the revenue (in dollars) from the sale of x units of a product is given byFind the marginal revenue when 49 units are sold. Interpret your result.
The revenue (in dollars) from the sale of x units of a product is given byFind the marginal revenue when 149 units are sold. Interpret your result.
A travel agency will plan a group tour for groups of size 25 or larger. If the group contains exactly 25 people, the cost is $300 per person. If each person's cost is reduced by $10 for each
McRobert's Electronics sells 200 TVs per month at a price of $400 per unit. Market research indicates that the store can sell one additional TV for each $1 it reduces the price. In this case the
1. The reaction R to an injection of a drug is related to the dosage x (in milligrams) according toR(x) = x2 (500 - x/3)where 1000 mg is the maximum dosage. If the rate of reaction with respect to
If a test having reliability r is lengthened by a factor n, the reliability of the new test is given byFind the rate at which R changes with respect to n.
The sales of a product s (in thousands of dollars) are related to advertising expenses (in thousands of dollars) bys = 200x / x + 10Find and interpret the meaning of the rate of change of sales with
Suppose that the proportion P of voters who recognize a candidate's name t months after the start of the campaign is given by(a) Find the rate of change of P when t = 6, and explain its meaning. (b)
Find the derivative, but do not simplify your answer.1. y = (7x6 - 5x4 + 2x2 - 1) (4x9 + 3x7 - 5x2 + 3x)2. y = (9x9 - 7x7 - 6x) (3x5 - 4x4 + 3x3 - 8)
It is determined that a wildlife refuge can support a group of up to 120 of a certain endangered species. If 75 are introduced onto the refuge and their population after t years is given byfind the
In January 2014 the so-called "Polar Vortex" of dense, frigid air plunged deep into the United States and resulted in record cold temperatures and dangerous wind chills. If s is the wind speed in
Experimental evidence has shown that the response y of a muscle is related to the concentration of injected adrenaline x according to the equation y = x / a + bx where a and b are constants. Find the
The table gives the number of millions of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030.With B(t) representing the number of beneficiaries (in
The table shows data for sulfur dioxide emissions from electricity generation (in millions of short tons) for selected years from 2000 and projected to 2035. These data can be modeled by the
For selected years from 1950 and with projections to 2050, the table shows the percent of total U.S. workers who were female.Assume these data can be modeled with the functionwhere p(t) is the
As the Baby Boom generation ages and the proportion of the U.S population over 65 increases, the number of Americans with Alzheimer's disease and other dementia is projected to grow each year. With
At each indicated point find (a) The slope of the tangent line, and (b) The instantaneous rate of change of the function. 1. y = (x2 + 1) (x3 - 4x) at (- 2, 0) 2. y = (x3 - 3) (x2 - 4x + 1) at (2, -
Find dy/du, du/dx, and dy/dx. 1. y = u3 and u = x2 + 1 2. y = u4 and u = x2 + 4x
At the indicated point, for each function, find(a) The slope of the tangent line,(b) The instantaneous rate of change of the function.You may use the numerical derivative feature on a graphing
Write the equation of the line tangent to the graph of each function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct. 1. y =
Complete the following for each function.(a) Find f(x).(b) Check your result in part (a) by graphing both it and the numerical derivative of the function.(c) Find x-values for which the slope of the
Do the following for each function f(x).(a) Find f'(x).(b) Graph both f(x) and f'(x) with a graphing utility.(c) Determine x-values where f'(x) = 0, f'(x) > 0, f'(x) < 0.(d) Determine x-values
1. Find the derivative of each function.(a) y = 2x3/3(b) y = 2/3x3(c) y = (2x)3/3(d) y = 2/(3x)32. (a) y = 3/(5x)5(b) y = 3x5/5(c) y = 3/5x5(d) y = (3x)5/5
Ballistics experts are able to identify the weapon that fired a certain bullet by studying the markings on the bullet. Tests are conducted by firing into a bale of paper. If the distance s, in
Suppose that the population of a certain microorganism at time t (in minutes) is given by P = 1000 - 1000(t + 10)-1 Find the rate of change of population.
The revenue from the sale of a product is R = 1500x + 3000(2x + 3)-1 - 1000 dollars where x is the number of units sold. Find the marginal revenue when 100 units are sold. Interpret your result.
The revenue from the sale of x units of a product is R = 15(3x + 1)-1 + 50x - 15 dollars Find the marginal revenue when 40 units are sold. Interpret your result.
Suppose that the weekly sales volume y (in thousands of units sold) depends on the price per unit (in dollars) of the product according to y = 32(3p + 1)-2/5, p > 0 (a) What is the rate of change in
A chain of auto service stations has found that its monthly sales volume S (in thousands of dollars) is related to the price p (in dollars) of an oil change according to(a) What is the rate of change
Suppose that the demand for q units of a product priced at $p per unit is described by p = 200,000 / (q + 1)2 (a) What is the rate of change of price with respect to the quantity demanded when q =
The relation between the magnitude of a sensation y and the magnitude of the stimulus x is given by y = k (x - x0)n where k is a constant, x0 is the threshold of effective stimulus, and n depends on
1. If the demand for q units of a product priced at $p per unit is described by the equationfind the rate of change of p with respect to q. 2. For the stimulus of electrical stimulation y = k(x -
The daily sales S (in thousands of dollars) attributed to an advertising campaign are given bywhere t is the number of weeks the campaign runs. What is the rate of change of sales at (a) t = 8? (b) t
Differentiate the functions. 5. f (x) = (3x5 - 2)20 6. g (x) = (3 - 2x)10 7. h(x) = 3/4 (x5 - 2x3 + 5)8
The data entry speed (in entries per minute) of a data clerk trainee isS = 10 √0.8x + 4, 0 ≤ x ≤ 100where x is the number of hours of training he has had. What is the rate at which his speed is

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