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calculus

Questions and Answers of Calculus

Prove that cosine is a continuous function.
For what values of is f continuous?
Is there a number that is exactly 1 more than its cube?
Show that the functionis continuous on (- ˆž, ˆž).
A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 PM. The following morning, he starts at 7:00 AM at the top and takes the same path
The toll charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 AM and 10 AM and between 4 PM and 7 PM) when the toll is $7. (a) Sketch a graph of as a
Guess the value of the limitby evaluating the function f(x) = x2/2x for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess.
Evaluate the limit and justify each step by indicating theappropriate properties of limits.
Find the limit or show that it does not exist.
For the function whose graph is given, state the following.(e) The equations of the asymptotes
(a) Estimate the value ofby graphing the function f(x) = ˆšx2 + x + 1 + x.(b) Use a table of values of f(x) to guess the value of the limit.(c) Prove that your guess is correct.
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.y = 2x + 1 / x - 2
Estimate the horizontal asymptote of the functionby graphing f for - 10 ‰¤ x ‰¤ 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain
Find a formula for a function that satisfies the following conditions:
Sketch the graph of an example of a function f that satisfies all of the given conditions.(a)(b) (c)
A function is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one -intercept, x = 1. It is known that has a removable discontinuity at x = -1 and limx†’-1 f(x) =
Find the limits as x → ∞ and as x → -∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.(a) y = x4 - x6(b) y = (3 - x) (1 + x)2 (1 - x)4
(b) Graph f(x) = (sin x)/x. How many times does the graph cross the asymptote?
Let P and Q be polynomials. Findif the degree of P is (a) less than the degree of Q and (b) greater than the degree of .
Find limx†’ˆž f(x) if, for all x > 1,
In Chapter 9 we will be able to show, under certain assumptions, that the velocity v(t) of a falling raindrop at time t isV(t) = v* (1 - e-gt/v*)where t is the acceleration due to gravity and v* is
Use a graph to find a number N such that If x > N then |3x2 + 1/2x2 + x + 1 -1.5| < 0.05
For the limitillustrate Definition 8 by finding values of N that correspond to Îµ = 0.5 and Îµ = 0.1.
(a) How large do we have to take so that 1/x2 (b) Taking r = 2 in Theorem 5, we have the statementProve this directly using Definition 7.
Use Definition 8 to prove that 1/x = 0.
Use Definition 9 to prove that ex = ˆž.
Prove thatandif 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.
(a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. When is the particle moving to the right? Moving to the left? Standing still?(b)
If a ball is thrown into the air with a velocity of 40 fts, its height (in feet) after seconds is given by y = 40t - 16 t2. Find the velocity when t = 2.
The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 1/t2, where is measured in seconds. Find the velocity of the particle at times t = a, t -
For the function t whose graph is given, arrange the following numbers in increasing order and explain your reasoning:0 g'(- 2) g'(0) g'(2) g'(4)
If an equation of the tangent line to the curve at the point where a =2 is y = 4x - 5, find f(2) and f'(2).
Sketch the graph of a function for which f(0) = 0, f'(0) = 3, f'(1) = 0, and f'(2) = -1.
If f(x) = 3x2 - x3, find f'(1) and use it to find an equation of the tangent line to the curve y = 3x2 - x3 at the point (1, 2).
(a) If F(x) = 5x/(1 + x2), find F'(2) and use it to find an equation of the tangent line to the curve y = 5x/ (1 + x2) at the point (2, 2). (b) Illustrate part (a) by graphing the curve and the
Find f'(a). (a) f(x) = 3x2 - 4x + 1 (b) f(t) = 2t + 1/t + 3 (c) f(x) = √1 - 2x
(a) Find the slope of the tangent line to the parabola y = 4x - x2 at the point (1, 3) (i) Using Definition 1 (ii) Using Equation 2 (b) Find an equation of the tangent line in part (a). (c) Graph the
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 and the speed when t = 5. F(t) = 100 + 50 t - 4.9 t2
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
The number N of US cellular phone subscribers (in millions) is shown in the table. (Midyear estimates are given.)(a) Find the average rate of cell phone growth (i) From 2002 to 2006 (ii) From 2002 to
The cost (in dollars) of producing units of a certain commodity is C(x) = 5000 + 10x + 0.05 x2.(a) Find the average rate of change of C with respect to when the production level is changed(i) From x
The cost of producing x ounces of gold from a new gold mine is C = f(x) dollars. (a) What is the meaning of the derivative f'(x)? What are its units? (b) What does the statement f'(800) = 17
Let T'(t) be the temperature (inoF) in Phoenix hours after midnight on September 10, 2008. The table shows values of this function recorded every two hours. What is the meaning of T'(8)? Estimate its
Find an equation of the tangent line to the curve at the given point. (a) y = 4x - 3x2, (2, - 4) (b) y = √x, (1, 1)
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
Determine whether f'(0) exists.
(a) Find the slope of the tangent to the curve y = 3 + 4x2 - 2x3 at the point where x = a. (b) Find equations of the tangent lines at the points (1, 5) and (2, 3). (c) Graph the curve and both
Use the given graph to estimate the value of each derivative. Then sketch the graph of f'.(a) f'(- 3)(b) f'(-2)(c) f'(- 1)(d) f'(0)(e) f'(1)(f) f'(2)(g) f'(3)
A rechargeable battery is plugged into a charger. The graph shows C(t), the percentage of full capacity that the battery reaches as a function of time elapsed (in hours).(a) What is the meaning of
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 4-11. Can you guess a formula for f'(x) from its graph? F(x) = ex
Let f(x) = x2. (a) Estimate the values of f'(0),f'(1/2), f'(1) and f'(2) by using a graphing device to zoom in on the graph of . (b) Use symmetry to deduce the values of f'(-1/2), f'(-1), and
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. a. F(x) = 1/2x - 1/3 b. f(t) = 5f - 9t2
Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV. Give reasons for your choices.(a)(b)(c)(d)IIIIIIIV
(a) If f(x) = x4 + 2x, find f'(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and f'.
The unemployment rate U(t) varies with time. The table (from the Bureau of Labor Statistics) gives the percentage of unemployed in the US labor force from 1999 to 2008.(a) What is the meaning of
The graph of f is given. State, with reasons, the numbers at which f is not differentiable.(a)(b)
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 points?
The figure shows the graphs of f, f, and f". Identify each curve, and explain your choices.
The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
Use the definition of a derivative to find f'(x) and f"(x). Then graph f, f', and f" on a common screen and check to see if your answers are reasonable. F(x) = 3x2 + 2x + 1
If f(x) = 2x2 - x3, find f'(x), f"(x), f"'(x), and f(4) x. Graph f, f', f"', and on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?
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.(a)(b) (c)
Let f(x) = 3√x. (a) If a ≠ 0, use Equation 2.7.5 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).
Show that the function f(x) = |x - 6| is not differentiable at 6. 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 is f differentiable? (c) Find a formula for f.
Recall that a function f is called even if f(-x) = f(x) for all 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
Let l be the tangent line to the parabola y = x2 at the point (1, 1). The angle of inclination of is the angle ϕ that l makes with the positive direction of the -axis. Calculate ϕ correct to the
The graph of f is given.(a) Find each limit, or explain why it does not exist.(b) State the equations of the horizontal asymptotes.(c) State the equations of the vertical asymptotes.(d) At what
If 2x - 1 ≤ f(x) ≤ x2 for 0 < x < 3, find limx→1 f(x).
Prove the statement using the precise definition of a limit.
Let(a) Evaluate each limit, if it exists.(b) Where is f discontinuous?(c) Sketch the graph of f.
Find the limit.
Show that the function is continuous on its domain. State the domain. h(x) = xesin x
Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval. x5 - x3 + 3x - 5 = 0, (1, 2)
(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.
The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + ¼ t2, where is measured in seconds. (a) Find the average velocity over each time period. (i) [1, 3] (ii)
(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 graphing
The total cost of repaying a student loan at an interest rate of r% per year is C = f(r). (a) What is the meaning of the derivative f'(r)? What are its units? (b) What does the statement f'(10) =
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.
(a) If f(x) = √3 - 5x, use the definition of a derivative to find f'(x). (b) Find the domains of f and f'. (c) Graph f and f' on a common screen. Compare the graphs to see whether your answer to
The graph of f is f shown. State, with reasons, the numbers at which f is not differentiable.
Let C(t) be the total value of US currency (coins and banknotes) in circulation at time . The table gives values of this function from 1980 to 2000, as of September 30, in billions of dollars.
Suppose that |f(x)| ≤ g(x) for all x, where limx→a g(X) = 0. Find limx→a f(x).
Evaluate 3ˆšx - 1 / ˆšx - 1.
(a) If we start from 0° latitude and proceed in a westerly direction, we can let T(x) denote the temperature at the point at any given time. Assuming that T is a continuous function of , show that
Suppose f is a function that satisfies the equationf(x + y) = f(x) + f(y) + x2 y + xy2for all real numbers x and y, suppose also that(a) Find f(0). (b) Find f'(0) (c) Find f'(x).
Find numbers a and b such that ˆšax + b - 2/ x = 1.
Evaluate the following limits, if they exist, where [x] denotes the greatest integer function.
Find all values of a such that f is continuous on R:
If limx→a [f(x) + g(x)] = 2 and limx→a [f(x) - g(x)] = 1, find limx→a [f(x) g(x)]
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
(a) What does it mean for f to be differentiable at a? (b) Sketch the graph of a function that is continuous but not differentiable at a = 2.
Describe several ways in which a limit can fail to exist. Illustrate with sketches.
Which of the following curves have vertical asymptotes?Which have horizontal asymptotes?(a) y = x4(b) y = sin x(c) y = tan x(d) y = tan-1 x(e) y = ex(f) y = ln x(g) y = 1/x(h) y = √x
(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?
(a) How is the number e defined?(b) Use a calculator to estimate the values of the limitsCorrect to two decimal places. What can you conclude about the value of e?
Differentiate the function. (a) f(x) = 240 (b) f(t) = 2 - 2/3t (c) f(x) = x3 - 4x + 6 (d) g(x) = x2 (1 - 2x)

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