Use the alternative form of the derivative to find the derivative at x = c, if it exists. g(x) = √│x│, c = 0
> Cuts in Cheese If you make the three complete cuts in the cheese as shown, how many pieces of cheese will you have?
> Is it possible for an argument to be invalid if the premises are all true? Explain your answer.
> In Exercises 35 and 36, (a) use lattice multiplication to perform the multiplication. (Hint: Be sure not to list any number greater than or equal to the base within the box.) Write the answer in the base in which the exercise is given. (b) Multiply the n
> Add in the indicated base.
> Add in the indicated base.
> Add in the indicated base.
> In a base 4 system, each of the four numerals is represented by one of the following colors: Determine the value of each color if the following addition is true in base 4.
> Determine b, by trial and error, if 1304b = 204.
> Consider the multiplication a) Multiply the numerals in base 8. b) Convert 4628 and 358 to base 10. c) Multiply the base 10 numerals determined in part (b). d) Convert the answer obtained in base 8 in part (a) to base 10. e) Are the answers obtai
> Perform the indicated operation
> Perform the indicated operation
> Write the Hindu–Arabic numerals in the numeration system discussed in Exercises 45–48 23
> Answer true or false. If false, give the reason. {circle} ( {square, circle, triangle}
> Reading a Map the scale on a map is 1 inch = 12 miles. How long a distance is a route on the map if it measures 4.25 in.?
> Use the rectangles in each graph to approximate the area of the region bounded by y = 5/x, y = 0, x = 1, and x = 5. Describe how you could continue this process to obtain a more accurate approximation of the area.
> Consider the function f(x) = 6x – x2 and the point P(2, 8) on the graph of f. a. Graph f and the secant lines passing through P(2, 8) and Q(x, f(x)) for x-values of 3, 2.5, and 1.5. b. Find the slope of each secant line. c. Use the results of part (b) to
> Consider the function f(x) = √x and the point P(4, 2) on the graph of f. a. Graph f and the secant lines passing through P(4, 2) and Q(x, f (x)) for x-values of 1, 3, and 5. b. Find the slope of each secant line. c. Use the results of part (b) to estimat
> A bicyclist is riding on a path modeled by the function f(x) = 0.08x, where x and f(x) are measured in miles (see figure). Find the rate of change of elevation at x = 2.
> A bicyclist is riding on a path modeled by the function f(x) = 0.04(8x − x2), where x and f(x) are measured in miles (see figure). Find the rate of change of elevation at x = 2.
> Decide whether the problem can be solved using pre-calculus or whether calculus is required. If the problem can be solved using pre-calculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical appr
> Decide whether the problem can be solved using pre-calculus or whether calculus is required. If the problem can be solved using pre-calculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical appr
> Discuss the relationship between secant lines through a fixed point and a corresponding tangent line at that fixed point.
> Describe the relationship between pre-calculus and calculus. List three pre-calculus concepts and their corresponding calculus counterparts.
> Use a graphing utility to graph the two functions f(x) = x2 + 1 and g(x) = │x│ + 1 in the same viewing window. Use the zoom and trace features to analyze the graphs near the point (0, 1). What do you observe? Which function is differentiable at this poin
> Find the slope of the tangent line to the graph of the function at the given point. f(t) = 3t − t2, (0, 0)
> Let and Show that f is continuous, but not differentiable, at x = 0. Show that g is differentiable at 0 and find g′(0).
> If a function is differentiable at a point, then it is continuous at that point.
> If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.
> If a function is continuous at a point, then it is differentiable at that point.
> If it is false, explain why or give an example that shows it is false. The slope of the tangent line to the differentiable function f at the point (2, f(2)) is f(2 + ∆x) - f (2) / ∆x.
> Consider the functions f(x) = x2 and g(x) = x3. a. Graph f and f′ on the same set of axes. b. Graph g and g′ on the same set of axes. c. Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h′(x)
> A line with slope m passes through the point (0, 4) and has the equation y = mx + 4. a. Write the distance d between the line and the point (3, 1) as a function of m. b. Use a graphing utility to graph the function d in part (a). Based on the graph, is t
> Determine whether the function is differentiable at x = 2.
> Determine whether the function is differentiable at x = 2.
> Find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = (1 − x)2/3
> Find the slope of the tangent line to the graph of the function at the given point. f(x) = 5 − x2 , (3, −4)
> Find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1?
> Find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = √1 – x2
> Find the derivatives from the left and from the right at x = 1 (if they exist). Is the function differentiable at x = 1? f(x) = │x − 1│
> Describe the x-values at which f is differentiable.
> Describe the x-values at which f is differentiable. f(x) = √x+1
> Describe the x-values at which f is differentiable.
> Describe the x-values at which f is differentiable. f(x) = (x + 4)2/3
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. f(x) = │x - 6│, c = 6
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. h(x) = │x + 7│, c = −7
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. g(x) = (x + 3)1/3 , c = −3
> Find the slope of the tangent line to the graph of the function at the given point. f(x) = 2x2 − 3, (2, 5)
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. f(x) = (x − 6)2-3 , c = 6
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. f(x) = 3/x, c = 4
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. g(x) = x2 − x, c = 1
> Use the alternative form of the derivative to find the derivative at x = c, if it exists. f(x) = x3 + 2x2 + 1, c = −2
> Evaluate f (2) and f (2.1) and use the results to approximate f′(2). f(x) = 1/4x3
> Evaluate f (2) and f (2.1) and use the results to approximate f′(2). f(x) = x(4 − x)
> Consider the function f(x) = 1/3 x3. a. Use a graphing utility to graph the function and estimate the values of f′(0), f′( 1/2), f′(1), f′(2) and f′(3). b. Use your results from part (a) to determine the values of f′(−1/2), f′(−1), f′(−2) and f′(−3). c
> Consider the function f(x) = 1/2 x2. a. Use a graphing utility to graph the function and estimate the values of f′(0), f′( 1/2), f′(1), and f′(2). b. Use your results from part (a) to determine the values of f′(−1/2), f′(−1), and f′(−2). c. Sketch a po
> The figure shows the graph of g′. a. g’(0) = b. g’(3) = c. What can you conclude about the graph of g knowing that g′(1) = - 8/3? d. What can you conclude about the graph of g kn
> Find the slope of the tangent line to the graph of the function at the given point. g(x) = 3/2 x + 1, (-2, -2)
> Use a graphing utility to graph each function and its tangent lines at x = −1, x = 0, and x = 1. Based on the results, determine whether the slopes of tangent lines to the graph of a function at different values of x are always distinct. a. f(x) = x2 b.
> Find equations of the two tangent lines to the graph of f that pass through the indicated point. f(x) = x2
> Find equations of the two tangent lines to the graph of f that pass through the indicated point. f(x) = 4x − x2
> Identify a function f that has the given characteristics. Then sketch the function. f(0) = 4; f′(0) = 0; f′(x) < 0 for x < 0; f′(x) > 0 for x > 0
> Identify a function f that has the given characteristics. Then sketch the function. f(0) = 2; f′(x) = −3 for −∞ < x < ∞
> The limit represents f′(c) for a function f and a number c. Find f and c.
> The limit represents f′(c) for a function f and a number c. Find f and c.
> The limit represents f′(c) for a function f and a number c. Find f and c.
> The limit represents f′(c) for a function f and a number c. Find f and c.
> The tangent line to the graph of y = h(x) at the point (−1, 4) passes through the point (3, 6). Find h(−1) and h′(−1).
> Find the slope of the tangent line to the graph of the function at the given point. f(x) = 3 − 5x, (-1, 8)
> The tangent line to the graph of y = g(x) at the point (4, 5) passes through the point (7, 0). Find g(4) and g′(4).
> A function f is symmetric with respect to the origin. Is f′ necessarily symmetric with respect to the origin? Explain.
> Do f and f′ always have the same domain? Explain.
> Sketch a graph of a function whose derivative is zero at exactly two points. Explain how you found the answer.
> Sketch a graph of a function whose derivative is always negative. Explain how you found the answer.
> Describe the relationship between continuity and differentiability.
> Describe how to find the derivative of a function using the limit process.
> List four notation alternatives to f′(x).
> Describe how to find the slope of the tangent line to the graph of a function at a point.
> Determine all polynomials P(x) such that P(x2 + 1) = (P(x))2 + 1 and P(0) = 0.
> Prove or disprove: If x and y are real numbers with y ≥ 0 and y(y + 1) ≤ (x + 1)2, then y(y − 1) ≤ x2.
> Sketch the graph of f′. Explain how you found your answer.
> a. Let f1(x) and f2(x) be continuous on the closed interval [a, b]. If f1(a) < f2(a) and f1(b) > f2(b), prove that there exists c between a and b such that f1(c) = f2(c). b. Show that there exists c in [0, π/2] such that cos x = x. Use a graphing utility
> Discuss the continuity of the function h(x) = x[[x]].
> Prove that if then f is continuous at c.
> Let What is the domain of f? How can you define f at x = 0 in order for f to be continuous there?
> Prove that for any real number y there exists x in (−π/2, π/2) such that tan x = y.
> Find all values of c such that f is continuous on (−∞, ∞).
> A swimmer crosses a pool of width b by swimming in a straight line from (0, 0) to (2b, b). (See figure.) a. Let f be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during t
> The table lists the frequency F (in Hertz) of a musical note at various times t (in seconds). a. Plot the data and connect the points with a curve. b. Does there appear to be a limiting frequency of the note? Explain.
> The signum function is defined by Sketch a graph of sgn(x) and find the following (if possible).
> Show that the function is continuous only at x = 0. (Assume that k is any nonzero real number.)
> Sketch the graph of f′. Explain how you found your answer.
> Show that the Dirichlet function is not continuous at any real number.
> Prove that if f is continuous and has no zeros on [a, b], then either f(x) > 0 for all x in [a, b] or f(x) < 0 for all x in [a, b].
> Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [5, 8], there is one with a volume of 1500 cubic centimeters.
> At 8:00 a.m. on Saturday, a man begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 a.m., he runs back down the mountain. It takes him 20 minutes to run up but only 10 minutes to run down. At some poin
> The number of units in inventory in a small company is given by where t is the time in months. Sketch the graph of this function and discuss its continuity. How often must this company replenish its inventory?
> A cell phone service charges $10 for the first gigabyte (GB) of data used per month and $7.50 for each additional gigabyte or fraction thereof. The cost of the data plan is given by C(t) = 10 − 7.5 [[1 – t]], t > 0 where t is the amount of data used (in
> Every day you dissolve 28 ounces of chlorine in a swimming pool. The graph shows the amount of chlorine f(t) in the pool after t days. Estimate and interpret / and /
> Describe how the functions f(x) = 3 + [[x]] and g(x) = 3 – [[-x]] differ.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The function f(x) = │x - 1│ / x – 1 is continuous on (-∞, ∞).
