I. Complete the diagram by filling in the boxes with the appropriate values. Show your work for finding δ given ε; a minimum of 3 lines of work needs to be shown for full credit. Find δ for ε = 0.005. Find L = (1 point)
A 50,000 -pound truck is parked on a 10∘ slope (see figure). Assume the only force to overcome is that due to gravity. Find the force required to keep the truck from rolling down the hill and the force perpendicular to the hill. (Round your answers to one decimal place.) Weight = 50,000 lb (a) Find the force required to keep the truck from rolling down the hill. lb (b) Find the force perpendicular to the hill. Ib
A toy wagon is pulled by exerting a force of 20 pounds on a handle that makes a 30∘ angle with the horizontal (see figure below). Find the work done in pulling the wagon 60 feet. (Round your answer to one decimal place.)
Find the area of the parallelogram that has the given vectors as adjacent sides. Use a computer algebra system or a graphing utility to verify your result. u = i + j + k v = j + k
Find the area of the triangle with the given vertices. (Hint: 12 ∥u×v∥ is the area of the triangle having u and v as adjacent sides.) A(0, 0, 0), B(4, 0, 6), C(−6, 3, 0)
The brakes on a bicycle are applied using a downward force of F = 18 pounds on the pedal when the crank makes a 40∘ angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. (Round your answer to two decimal places.) ft-lb
A force of 24 pounds acts on the pipe wrench shown in the figure below. (a) Find the magnitude of the moment about O by evaluating ∥OA→×F∥. (0 ≤ θ ≤ 180∘) Use a graphing utility to graph the resulting function of θ. (b) Use the result of part (a) to determine the magnitude of the moment when θ = 45∘. (Round your answer to two decimal places.)
The Heaviside function H is defined by H(t) = {0 if t < 0 1 if t ≥ 0. It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and 150 volts are applied instantaneously to the circuit Write a formula for V(t) in terms of H(t). V(t) = 150H(t) V(t) = H(t) 150 V(t) = 150 V(t) = H(t) + 150 V(t) = H(t) − 150 (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 8 seconds and 270 volts are applied instantaneously to the circuit. Write a formula for V(t) in terms of H(t). (Note that starting at t = 8 corresponds to a translation.) V(t) = 270 H(t + 8) V(t) = 270H(t) − 8 V(t) = 8H(t + 270) V(t) = 8H(t − 270) V(t) = 270H(t − 8)
The Heaviside function H is defined below. It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. H(t) = {0 if t < 0 1 if t ≥ 0 The Heaviside function can also be used to define the ramp function y = ctH(t), which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y = tH(t). (b) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 0 and the voltage is gradually increased to 140 volts over a 70 -second time interval. Write a formula for V(t) in terms of H(t) for t ≤ 70. V(t) = 2H(t) − t V(t) = 2tH(t) V(t) = H(t) − 2t V(t) = H(t)−t 2 V(t) = 1 2 − tH(t) (c) Sketch the graph of the voltage V(t) in a circuit if the switch is turned on at time t = 8 seconds and the voltage is gradually increased to 220 volts over a period of 55 seconds. Write a formula for V(t) in terms of H(t) for t ≤ 63. V(t) = 4H(t − 8) V(t) = 4(t−8)H(t−8) V(t) = 4(t+8)H(t+8) V(t) = 1 4 −(t−8)H(t−8) V(t) = 1 4 − (t+8)H(t+8)
A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 10 cm/s. (a) Express the radius r of the balloon as a function of the time t (in seconds). r(t) = (b) If V is the volume of the balloon as a function of the radius, find V∘r. (V∘r)(t) = Interpret the answer found in part (b). This formula gives the volume of the balloon (in cm3) as a function of time (in seconds). This formula gives the amount of time (in seconds) the balloon has been inflating as a function of V.
27-28 Find an explicit formula for f−1 and use it to graph f−1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f−1 are reflections about the line. 27. f(x) = x4 + 1, x ⩾ 0 28. f(x) = 2 − ex
31-32 Find an explicit formula for f−1 and use it to graph f−1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f−1 are reflections about the line. 31. f(x) = 4x + 3 32. f(x) = 1 + e−x
An airplane is flying at a speed of 300 mi/h at an altitude of one mile and passes directly over a radar station at time t = 0. (a) Express the horizontal distance d (in miles) that the plane has flown as a function of t. d(t) = (b) Express the distance s between the plane and the radar station as a function of d. s(d) = (c) Use composition to express s as a function of t. (s∘d)(t) =
The graph of f is given. (a) Why is f one-to-one? f is one-to-one because it passes the (b) What are the domain and range of f−1 ? (Enter your answers in interval notation.) domain range (c) What is the value of f−1(1) ? (d) Estimate the value of f−1(0) to the nearest tenth.
A ship is moving at a speed of 40 km/h parallel to a straight shoreline. The ship is 8 km from shore, and it passes a lighthouse at noon. (a) Express the distance s (in km ) between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s = f(d). s = f(d) = km (b) Express d (in km ) as a function of t (in hours), the time elapsed since noon; that is, find g so that d = g(t). d = g(t) = km (c) Find f∘g. (f∘g)(t) = What does this function represent? This function represents the distance between the lighthouse and the ship as a function of the time elapsed since noon. This function represents time elapsed since noon as a function of the distance between the lighthouse and the ship.
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 50 cm/s. (a) Express the radius r (in cm ) of this circle as a function of time t (in seconds). r(t) = cm (b) If A is the area of this circle as a function of the radius, find A∘r. (A∘r)(t) = Interpret your answer. This formula gives the extent of the rippled area (in cm2) at any time t. This formula gives the circumference of the rippled area (in cm ) at any time t. This formula gives the radius of the rippled area (in cm) at any time t.
(a) How is the logarithmic function y = logb(x) defined? It is defined as the inverse of the exponential function with base b, that is, logb(x) = y ⇔ x = . (b) What is the domain of this function? (Enter your answer using interval notation.) (c) What is the range of this function? (Enter your answer using interval notation.) (d) Sketch the general shape of the graph of the function y = logb(x) if b > 1.
(a) What is a one-to-one function? A function f is called a one-to-one function if it periodically takes on the same value. A function f is called a one-to-one function if it has negative slope. A function f is called a one-to-one function if it never takes on the same value twice. A function f is called a one-to-one function if it is a vertical line. A function f is called a one-to-one function if it is a horizontal line. (b) How can you tell from the graph of a function whether it is one-to-one? It must pass the Vertical Line Test. It must always increase in value. You cannot tell by looking at the graph. It must always decrease in value. It must pass the Horizontal Line Test.
A function is given by a table of values. Determine whether it is one-to-one. Yes, it is one-to-one. No, it is not one-to-one. A function is given by a table of values. Determine whether it is one-to-one. Yes, it is one-to-one. No, it is not one-to-one.
A function is given by a graph. Determine whether it is one-to-one. Yes No A function is given by a graph. Determine whether it is one-to-one. Yes, it is one-to-one. No, it is not one-to-one. A function is given by a graph. Determine whether it is one-to-one. Yes, it is one-to-one. No, it is not one-to-one. A function is given by a graph. Determine whether it is one-to-one. Yes, it is one-to-one. No, it is not one-to-one.
A function is given by a formula. Determine whether it is one-to-one. f(x) = x2 − 12 Yes, it is one-to-one. No, it is not one-to-one. A function is given by a formula. Determine whether it is one-to-one. g(x) = 1 − sin(x) Yes, it is one-to-one. No, it is not one-to-one.
A function is given by a verbal description. Determine whether it is one-to-one. The function f(t) is the height of a football t seconds after kickoff. Yes, it is one-to-one. No, it is not one-to-one. A function is given by a verbal description. Determine whether it is one-to-one. The function f(t) is your height at age t Yes, it is one-to-one. No, it is not one-to-one.