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mathematics
calculus

Questions and Answers of Calculus

Formulate a precise definition of lim x→∞ f(x) = -∞. Then use your definition to prove that lim x→∞ (1 + x3) = - ∞
Prove that lim x→∞ f(x) = lim x→0+ f(1/t) and lim x→∞ f(x) = lim x→0- f(1/t) if these limits exist.
A curve has equation y = f(x). (a) Write an expression for the slope of the secant line through the points P(3, f(3)) and Q(x, f(x)).(b) Write an expression for the slope of the tangent line at P.
Suppose an object moves with position function s = f(t).(a) Write an expression for the average velocity of the object in the time interval from t = a to t = a + h.(b) Write an expression for the
Consider the slope of the given curve at each of the five points shown. List these five slopes in decreasing order and explain your reasoning.
Graph the curve y = ex in the viewing rectangles [– 1, 1] by [0, 2] [– 0.5, 0.5], by [– 0.5, 0.5], and [– 1, 0.1] by. What do you notice about the curve as you zoom in toward the point (0, 1)?
(a) Find the slope of the tangent line to the parabola y = x2 + 2x at the point (– 3, 3) (i) Using Definition 1(ii) Using Equation 2(b) Find an equation of the tangent line in part (a).(c) Graph
(a) Find the slope of the tangent line to the curve y = x3 at the point (– 1, – 1) (i) Using Definition 1(ii) Using Equation 2(b) Find an equation of the tangent line in part (a).(c) Graph the
Find an equation of the tangent line to the curve at the given point.
(a) Find the slope of the tangent to the curve y = 2/(x + 3) at the point where x = a.(b) Find the slopes of the tangent lines at the points whose x-coordinates are (i) , (ii) 0, and (iii) – 1.
(a) Find the slope of the tangent to the parabola y = 1 + x + x2 at the point where x = a. (b) Find the slopes of the tangent lines at the points whose -coordinates are (i), (ii), and (iii) 1. (c)
(a) Find the slope of the tangent to the curve y = x3 – 4x + 1 at the point where x = a.(b) Find equations of the tangent lines at the points (1 – 2) and (2, 1).(c) Graph the curve and both
(a) Find the slope of the tangent to the curve y = 1/√x at the point where x = a. (b) Find equations of the tangent lines at the points (1, 1) and (4, ½); (c) Graph the curve and both
The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions.(a) What was the initial velocity of the car?(b) Was the car going faster
Valerie is driving along a highway. Sketch the graph of the position function of her car if she drives in the following manner: At time t = 0, the car is at mile marker 15 and is traveling at a
If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after seconds is given by Y = 40t – 16t2. Find the velocity when t = 2.
If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after seconds is given by H = 58t – 0.83t2.(a) Find the velocity of the arrow after one second.(b) Find the
The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 4t3 + 6t + 2, where is measured in seconds. Find the velocity of the particle at times t =
The displacement (in meters) of a particle moving in a straight line is given by s = t2 – 8t + 18, where is measured in seconds.(a) Find the average velocity over each time interval:(i) [3,
A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate
A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. The graph shows how the temperature of the turkey
(a) Use the data in Example 5 to find the average rate of change of temperature with respect to time(i) From 8 P.M. to 11 P.M.(ii) From 8 P.M. to 10 P.M.(iii) From 8 P.M. to 9 P.M.(b) Estimate the
The population P (in thousands) of Belgium from 1992 to 2000 is shown in the table. (Midyear estimates are given.)(a) Find the average rate of growth(i) From 1992 to 1996(ii) From 1994 to 1996(iii)
The number N (in thousands) of cellular phone subscribers in Malaysia is shown in the table. (Midyear estimates are given.)(a) Find the average rate of growth(i) From 1995 to 1997(ii) From 1995 to
The number N of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of June 30 are given.)(a) Find the average rate of growth(i) From 1996 to 1998(ii) From
The cost (in dollars) of producing units of a certain commodity is C(x) = 5000 + 10x + 0.05x2.(a) Find the average rate of change of C with respect to when the production level is changed(i) From x =
If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli€™s Law gives the volume V of water remaining in the tank
On the given graph of f, mark lengths that represent f(2), f(2 + h), f(2 + h) €“ f(2), and h. (choose h > 0,). What line has slope f(2 + h) €“ f(2) / h?
For the function f whose graph is shown in Exercise 1, arrange the following numbers in increasing order and explain your reasoning:0 f ’(2) f ’(3) - f ’(2) ½[ f ’(4) - f (2)]
For the function t whose graph is given, arrange the following numbers in increasing order and explain your reasoning:
If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f (4) and f’ (4).
Sketch the graph of a function f for which f(0) = 0, f’(0) = 3, f’(1) = 0, and f’(2) = - 1.
Sketch the graph of a function f for which f(0) = 0, f’(0) = 3, g’(1) = 0, and g’(2) = - 1.
If f (x) = 3x2 – 5x, find f’(2) and use it to find an equation of the tangent line to the parabola y = 3x2 – 5x at the point (2, 2).
If g(x) = 1 – x3, find g’ (0) and use it to fine an equation of the tangent line to the curve y = 1 – x3 at the point (0, 1).
(a) If F(x) = x3 – 5x + 1, find F’ (1) and use it to find an equation of the tangent line to the curve y = x3 – 5x + 1 at the point (1, - 3).(b) Illustrate part (a) by graphing the curve and
(a) If G(x) = x/(1 + 2x), find G’(a) and use it to fine an equation of the tangent line to the curve y = x/(1 + 2x) at the point (- ¼, - ½).(b) Illustrate part (a) by graphing the curve and the
Let f (x) = 3*. Estimate the value of f’(1) in two ways;(a) By using Definition 2 and taking successively smaller values of h.(b) By zooming in on the graph of y = 3* and estimating the slope.
Let g(x) = tan x. Estimate the value of g’(π/4) in two ways; (a) By using Definition 2 and taking successively smaller values of h. (b) By zooming in on the graph of y = tan x and estimating
Find f€™ (a)
Each limit represents the derivative of some function f at some number a. State such an f and in each case.
A particle moves along a straight line with equation of motion s = f(t), where is measured in meters and in seconds. Find the velocity when t = 2.
The cost of producing x ounces of gold from a new gold mine C = f(x) is dollars.(a) What is the meaning of the derivative f ’(x)? What are its units?(b) What does the statement f’ (800) = 17
The number of bacteria after t hours in a controlled laboratory experiment is n = f(t).(a) What is the meaning of the derivative f’(5)? What are its units?(b) Suppose there is an unlimited amount
The fuel consumption (measured in gallons per hour) of a car traveling at a speed of miles per hour is c = f (v).(a) What is the meaning of the derivative f’(v)? What are its units?(b) Write a
The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f (p)..(a) What is the meaning of the derivative f’(8)? What are its
Let be the temperature (in oF) in Dallas hours after midnight on June 2, 2001. The table shows values of this function recorded every two hours. What is the meaning of T€™ (10)? Estimate
Life expectancy improved dramatically in the 20th century. The table gives values of E(t), the life expectancy at birth (in years) of a male born in the year t in the United States. Interpret and
The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water) The graph shows how oxygen solubility varies
The graph shows the influence of the temperature T on the maximum sustainable swimming speed of Coho salmon.(a) What is the meaning of the derivative S'(T)? What are its units?(b) Estimate the values
Determine whether f€™ (0) exists?
Use the given graph to estimate the value of each derivative. Then sketch the graph of f.
Match the graph of each function in (a)€“(d) with the graph of its derivative in I€“IV. Give reasons for your choices.
Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f below it.
Shown is the graph of the population function P(t) for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative P'(t). What does the graph of P' tell us about the
The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M'(t). During which years was the
Make a careful sketch of the graph of f and below it sketch the graph of f€™ in the same manner as in Exercises 5€“13. Can you guess a formula for f €™(x) from its
Let f(x) = x2.(a) Estimate the values of f’ (0),f’(½), f’(1), and f’(2)by using a graphing device to zoom in on the graph of f.(b) Use symmetry to deduce the values of f’ (-1/2), f’
Let f(x) = x3.(a) Estimate the values of f’ (0), f’(½), f’(1), and f’(2) and f’ (3) by using a graphing device to zoom in on the graph of f.(b) Use symmetry to deduce the values of f’
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
(a) Sketch the graph of f(x) = √6 – x by starting with the graph of y = √x and using the transformations of Section 1.3. (b) Use the graph from part (a) to sketch the graph of
(a)If f(x) = x – (2/x), find f’(x).(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f’.
(a)If f(t) = 6/ (1 + t2), find f’(t).(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f’.
(a) What is the meaning of U€™(t)? What are its units?(b) Construct a table of values for U€™ (t)
Let P(t) be the percentage of Americans under the age of 18 at 3 x time . The table gives values of this function in census years from 1950 to 2000.(a) What is the meaning of P€™ (t)? What
The graph of f is given. State, with reasons, the numbers at which f is not differentiable.
The graph of is given.(a) At what numbers is discontinuous? Why?(b) At what numbers is not differentiable? Why?
Graph the function f(x) = x + √| x |. Zoom in repeatedly, first toward the point (-1, 0) and then toward the origin. What is different about the behavior of f in the vicinity of these two
Zoom in toward the points (1, 0), (0, 1), and (-1, 0) on the graph of the function g(x) = (x2 – 1)2/3. What do you notice? Account for what you see in terms of the differentiability of g.
Let f(x) = 3√x. (a) If a ≠ 0, use Equation 2.8.3 to find f’(a). (b) Show that f’(0) does not exist. (c) Show that y = 3√x has a vertical tangent line at (0, 0). (Recall the
(a) If g(x) = x2/3, show that g’ (0) does not exist. (b) If a ≠ 0 find g’ (0) (c) Show that y = x2/3 has a vertical tangent line at (0, 0). (d) Illustrate part (c) by graphing y = x2/3.
Show that the function f(x) = | x – 6 | is not differentiable at 6. Find a formula for f’ and sketch its graph.
Where is the greatest integer function f(x) = [x]not differentiable? Find a formula for f’ and sketch its graph.
(a) Sketch the graph of the function f(x) = x | x |.(b) For what values of x is f differentiable?(c) Find a formula for f’.
The left-hand and right-hand derivatives of f at are defined by if these limits exist. Then f'(a) exists if and only if these one sided derivatives exist and are equal.(a) Find f'-(4) and f'+(4) for
Recall that a function is f called even if f(-x) = f(x) for all x in its domain and odd if f(-x) = - f(x)for all such x. Prove each of the following.(a) The derivative of an even function is an odd
When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running.(a) Sketch a possible graph of T as a function of the time that has elapsed since
Let e be the tangent line to the parabola y = x2 at the point (1, 1). The angle of inclination of is the angle Ф that makes with the positive direction of the -axis. Calculate Ф correct
Explain what each of the following means and illustrate with a sketch.
Which of the following curves have vertical asymptotes? Which have horizontal asymptotes?
(a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval (– ∞, ∞)? What can you say about the graph of such a function?
If y = f (x) and x changes from x1 to x2, write expressions for the following.(a) The average rate of change of y with respect to x over the interval [x1, x2](b) The instantaneous rate of change of y
Define the derivative f’(a). Discuss two ways of interpreting this number.
(a) What does it mean for f to be differentiable at a?(b) What is the relation between the differentiability and continuity of a function?(c) Sketch the graph of a function that is continuous but not
The graph of f is given.(i) lim x→2+ f(x) (ii) lim x→3+ f(x)(iii) lim x→€“2 f(x) (iv) lim x→4 f(x)(v) lim x→0 f(x) (vi) lim x→2€“ f(x)(vii) lim
Sketch the graph of an example of a function f that satisfies all of the following conditions:
Find the limit.
Use graphs to discover the asymptotes of the curve. Then prove what you have discovered.
If 2x – 1 < f(x) x2 for 0 < x < 3, find lim x→1 f(x)
Prove that lim x→0 x2 cos (1/x2) = 0.
Prove the statement using the precise definition of a limit.
Let(a) Evaluate each limit, if it exists.(i) lim x→0+ f (ii) lim x→0- f (iii) lim x→0 f(iv) lim x→3- f (v) lim x→3+ f (vi) lim x→3 f(b) Where is f
(a) For each of the numbers 2, 3, and 4, discover whether is continuous from the left, continuous from the right, or continuous at the number.(b) Sketch the graph of g.
Show that the function is continuous on its domain. State the domain.
Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval.
(a) Find the slope of the tangent line to the curve y = 9 – 2x2 at the point (2, 1).(b) Find an equation of this tangent line.
Find equations of the tangent lines to the curve y = 2/1 – 3x at the points with -coordinates and
The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + t2/4, where is measured in seconds.(a) Find the average velocity over each time period.(i) [1, 3] (ii)
According to Boyle’s Law, if the temperature of a confined gas is held fixed, then the product of the pressure P and the volume is a constant. Suppose that, for a certain gas, PV = 800, where P is
(a) Use the definition of a derivative to find f’ (2), where f(x) = x3 – 2x.(b) Find an equation of the tangent line to the curve y = x3 – 2x at the point (2, 4).(c) Illustrate part (b) by

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