The graph of f is shown. Apply Rolle's Theorem and find all values of c such that f′(c) = 0 at some point between the labeled intercepts. (Enter your answers as a comma-separated list.) f(x) = sin(2x)
The graph of f is shown. Apply Rolle's Theorem and find all values of c such that f′(c) = 0 at some point between the labeled intercepts. (Enter your answers as a comma-separated list.) f(x) = x2 + 4x − 21
The graph of f is shown. Apply Rolle's Theorem and find all values of c such that f′(c) = 0 at some point between the labeled intercepts. (Enter your answers as a comma-separated list.) f(x) = −sin(2x) c =
Find the two x-intercepts of the function f and show that f′(x) = 0 at some point between the two x-intercepts. f(x) = x2 − 4x − 12 (x, y) = ( ) (smaller x-value) (x, y) = ( ) (larger x-value) Find a value of x such that f′(x) = 0. x =
The graph of f is shown. Apply Rolle's Theorem and find all values of c such that f′(c) = 0 at some point between the labeled intercepts. (Enter your answers as a comma-separated list.) f(x) = 2sin2x c =
Find the two x-intercepts of the function f and show that f′(x) = 0 at some point between the two x intercepts. f(x) = x x+3 (x, y) = ( ) (smaller x-value) (x, y) = ( ) (larger x-value) Find a value of x such that f′(x) = 0. x =
Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = −x2 + 7x, [0, 7] Yes, Rolle's theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = x4/5 + 7, [−32, 32] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a)≠f(b). If Rolle's theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = x2 − 4x − 21 x + 6, [−3, 7] Yes. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b) If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = sin(x), [0, π] Yes. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b) If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = tan(x), [−2π, −π] Yes. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) ≠ f(b) If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. (Enter your answers as a comma-separated list. If Rolle's Theorem cannot be applied, enter NA.) c =
Use a graphing utility to graph the function on the closed interval [a, b]. Determine whether Rolle's Theorem can be applied to f on the interval and, if so, find all values of c in the open interval (a, b) such that f′(c) = 0. f(x) = 9x − tan(πx), [−1/4, 1/4]
The height of a ball t seconds after it is thrown upward from a height of 6 feet and with an initial velocity of 80 feet per second is f(t) = −16t2 + 80t + 6. (a) Verify that f(2) = f(3). f(2) = ft f(3) = ft (b) According to Rolle's Theorem, what must be the velocity at some time in the interval (2, 3) ? ft/sec Find that time. t = s
Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = x2, [4, 5] Yes, the Mean Value Theorem can be applied. No, f is not continuous on [a, b]. No, f is not differentiable on (a, b). None of the above. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = f(b) − f(a) b − a. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.) c =
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = f(b) − f(a) b − a. If the Mean Value Theorem cannot be applied, explain why not. f(x) = x4/5, [0, 1]
Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = |3x + 7|, [−5, 1] Yes, the Mean Value Theorem can be applied. No, f is not continuous on [a, b]. No, f is not differentiable on (a, b). None of the above. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = f(b) − f(a) b − a. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.) c =
Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply. ) f(x) = (x + 3) ln(x + 3), [−2, −1] Yes, the Mean Value Theorem can be applied. No, f is not continuous on [a, b]. No, f is not differentiable on (a, b). None of the above. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = f(b) − f(a) b − a. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.) c =
Find the slope of the tangent line to the graph of the function at the point (0, 1). (a) y = e4x (b) y = e−4x
Find an equation of the tangent line to the graph of the logarithmic function at the point (1, 0). y = ln x8
Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. (If an answer is undefined, enter UNDEFINED.) f(t) = 4t + 2 t − 5 f′(6) = (6, 26)
Find F as a function of x and evaluate it at x = 5, x = 7, and x = 9. F(x) = ∫5 x −2 t3 dt F(x) = F(5) = F(7) = F(9) =
Find F as a function of x and evaluate it at x = 4, x = 7, and x = 10. (Round your answers to four decimal places.) F(x) = ∫1 x cos(θ)dθ F(x) = F(4) = (7) = F(10) =
Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Enter your answers as a comma-separated list.) f(x) = x9, [0, 9] c =
Find the area of the region bounded by the graphs of the equations. Use a graphing utility to verify your result. (Round your answer to three decimal places.) y = x2+6 x, x = 1, x = 2, y = 0
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let n = 4 and round your answers to four decimal places. Use a graphing utility to verify your result. ∫2 6 lnx dx using the Trapezoidal Rule using Simpson's Rule calculator approximation
A particle is moving along the x-axis. The position of the particle at time t is given by x(t) = t3 − 6t2 + 9t − 2, 0 ≤ t ≤ 7. Find the total distance the particle travels in 7 units of time. units
Find the area of the region bounded by the graphs of the equations. y = −x2 + 6x, y = 0
Use a graphing utility to graph the integrand. ∫0 π sinx dx Use the graph to determine whether the definite integral is positive, negative, or zero. positive negative zero
Find F as a function of x and evaluate it at x = 4, x = 6, and x = 8. F(x) = ∫4 x −2 t3 dt
Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. f(x) = 4x3 − 3x2, [−2, 4]
Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. f(x) = cosx, [−π/3, π/3]
Consider the following. Use the Vertical Line Test to determine whether the curve is the graph of a function of x. Yes, the curve is a function of x. No, the curve is not a function of x. If the curve is a function, state the domain and range. (Enter your answers using interval notation. If the curve is not a function enter NONE.) domain range
In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $13,000? On $25,000? on $13,000 $ on $25,000 $
A spherical balloon with radius r inches has volume V(r) = 4 3 πr3. Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 3 inches. (Express your answer in terms of π and r.)
Evaluate f(−5), f(0), and f(3) for the piecewise defined function. f(x) = {x − 5 if x ≤ −1 x2 if x > −1 f(−5) = f(0) = f(3) = Sketch a graph of the function.
Find an expression for the function whose graph is the given curve. The bottom half of the parabola x + (y − 1)2 = 0 y =
Find an expression for the function whose graph is the given curve. (Assume that the points are in the form (x, f(x)).) The line segment joining the points (3, −4), and (7, 2) f(x) = Find the domain of the function. (Enter your answer using interval notation.)
Recent studies indicate that the average surface temperature of a planet has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.01t + 8.75, where T is temperature in ∘C and t represents years since 1900. (a) What do the slope and T-intercept represent? The slope is, which means that the average surface temperature of the planet is, which presents the average surface temperature in of ∘C per year. The T-intercept is the year. (b) Use the equation to predict the planet's average surface temperature in 2010. ∘C
Question 3 A spherical balloon with radius r inches has a volume V(r) = 4 3 πr3. (a) Find an expression for the amount of air required to inflate the balloon so that the radius increases from r to r+3. (b) Find an expression for the amount of air required to double the radius of the balloon from a radius of r to a radius of 2r. (c) If the balloon starts with a volume of 36π cubic inches and the radius is increasing at 1 inch per minute, then how long will it take for the balloon to inflate to a volume of 200 cubic inches?
The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 AM? MW At 9 PM? MW (b) When was the power consumption the lowest? 6 PM midnight noon 4 AM 2 PM When was it the highest? midnight 6 PM noon 4 AM 2 PM Do these times seem reasonable? Yes No
Consider the following graph. Determine whether the curve is the graph of a function of x. Yes, it is a function. No, it is not a function. If it is, state the domain and range of the function. (Enter your answers using interval notation. If it is not a function, enter DNE in all blanks.) domain range
The graph of a function f is given in the figure. (a) Find the value of f(1). (b) Estimate the value of f(−1). (c) For what values of x is f(x) = 1 ? (Enter your answers as a comma-separated list.) (d) Estimate the value of x such that f(x) = 0. x = (e) State the domain and range of f. (Enter your answers in interval notation.) domain range (f) On what interval is f increasing? (Enter your answer using interval notation.)
A cell phone plan has a basic charge of $40 a month. The plan includes 500 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C (in dollars) as a function of the number x of minutes used. C(x) = { if ≤ x ≤ if x > Graph C as a function of x for 0 ≤ x ≤ 750.
Find a formula for the described function. Express the area of an equilateral triangle as a function of the length of a side x. A(x) = State the domain of A. (Enter your answer using interval notation.)
Consider the following graph. Determine whether the curve is the graph of a function of x. Yes, it is a function. No, it is not a function. If it is, state the domain and range of the function. (Enter your answers in interval notation. If the curve is not the graph of a function of x, enter DNE.) range
A spherical balloon is inflating with helium at a rate of 64π ft3 min. How fast is the balloon's radius increasing at the instant the radius is 2 ft? Write an equation relating the volume of a sphere, V, and the radius of the sphere, r. (Type an exact answer, using π as needed.) Differentiate both sides of the equation with respect to t. dV dt = ( ) dr dt (Type an exact answer, using π as needed. Type an expression using r as the variable.) The balloon's radius is increasing at a rate of at the instant the radius is 2 ft. (Simplify your answer.)
Evaluate f(−7), f(0), and f(6) for the piecewise defined function. f(x) = {x + 4 if x < 0 3 − x if x ≥ 0 f(−7) = f(0) = f(6) = Sketch the graph of the function.
In a certain state the maximum speed permitted on freeways is 65 mi/h, and the minimum is 40 mi/h. The fine F for violating these limits is $50 for every mile above the maximum or below the minimum. (a) Complete the expressions in the following piecewise defined function, where x is the speed (in mi/h) at which you are driving. F(x) = {if 0 < x < 40 if 40 ≤ x ≤ 65 if x > 65 (b) Find F(20), F(45), and F(85). F(20) = F(45) = F(85) = (c) What do your answers in part (b) represent? The awards for obeying the speed limits on the freeway. The maximum speed one can drive without paying a fine. The fines for violating the speed limits on the freeway. The initial speed at which a fine will be incurred. The minimum speed one can drive without paying a fine.
The graph shows the power consumption for a day in September in San Francisco where P is measured in megawatts and t is measured in hours starting at midnight. (a) What was the power consumption (in MW) at 3 a.m.? MW What was the power consumption (in MW) at 9 p.m.? MW (b) When was the power consumption the lowest? 2 p.m. 4 a.m. midnight 6 p.m. noon When was it the highest? midnight 2 p.m. noon 4 a.m. 6 p.m.
A spherical balloon with radius r inches is being inflated. The function V whose graph is sketched in the figure gives the volume of the balloon, V(t), measured in cubic inches after t seconds. At what approximate rate is the radius of the balloon changing after 4 seconds?
function of the driving speed x. Graph F for 0 ≤ x ≤ 100. In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 mi/h. The fine for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph F(x) for 0 ⩽ x ⩽ 100.
Find a function whose graph is the given curve. the bottom half of the circle x2 + y2 = 16 f(x) =
Find a formula for the described function. An open rectangular box with volume 7 m3 has a square base. Express the surface area SA of the box as a function of the length of a side of the base, x. SA = m2 State the domain of SA. (Enter your answer in interval notation.)
Consider the following graph. (i) Determine whether the curve is the graph of a function of x. Yes, it is a function. No, it is not a function. If it is, state the domain and range of the function. (Enter your answers using interval notation. If it is not a function, enter NAF in all blanks.) domain range
1. The radius of a spherical balloon is increasing at a rate of 4 centimeters per minute. How fast is the surface area changing when the radius is 14 centimeters? Hint: The surface area is S = 4πr2. Rate of change of surface area = cm /min 2. The surface area of a spherical balloon is increasing at a rate of 22 centimeters squared per minute. How fast is the radius changing when the radius is 14 centimeters? Rate of change of radius = cm/min
The graph of a function f is given. (a) State the value of f(−1). f(−1) = (b) Estimate the value of f(2). f(2) = (c) For what values of x is f(x) = 2? (Enter your answers as a comma-separated list.) x = (d) Estimate the values of x such that f(x) = 0. (Enter your answers as a comma-separated list. ) x = (e) State the domain and range of f. (Enter your answer using interval notation.) domain range
Use the Vertical Line Test to determine whether the curve is the graph of a function of x. Yes, the curve is a function of x. No, the curve is not a function of x. If the curve is a function, state the domain and range. (Enter your answers using interval notation. If the curve is not a function enter NONE.) range
Shown is a graph of the global average temperature T during the 20th century. (a) What was the global average temperature in 2000? (b) In what year was the average temperature 14.2∘? (c) When was the temperature smallest? When was the temperature largest? (d) Estimate the range of T. (Enter your answer using interval notation.)
Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? runner A runner B runner C Did each runner finish the race? Yes No
Sketch a rough graph of the number of hours of daylight (in the northern hemisphere) as a function of the time of year.
Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
Graphs of f and g are shown. Is f even, odd, or neither? even odd neither Explain your reasoning. It is symmetric about the origin. It is symmetric with respect to the y-axis. It is symmetric with respect to the x-axis. It is not symmetric about the origin or the y-axis. Is g even, odd, or neither? even odd neither Explain your reasoning. It is symmetric about the origin. It is symmetric with respect to the y-axis. It is symmetric with respect to the x-axis. It is not symmetric about the origin or the y-axis.
Graphs of f and g are shown. Is f even, odd, or neither? even odd neither Explain your reasoning. It is symmetric about the origin. It is symmetric with respect to the y-axis. It is symmetric with respect to the x-axis. It is not symmetric about the origin or the y-axis. Is g even, odd, or neither? even odd neither Explain your reasoning. It is symmetric about the origin. It is symmetric with respect to the y-axis. It is symmetric with respect to the x-axis. It is not symmetric about the origin or the y-axis.
Consider the differential equation y′ = y y2−1 (a) State the order of the differential equation . Then decide if the differential equation is linear or nonlinear. (b) Prove using an appropriate Existence and Uniqueness Theorem that every IVP of the form y(a) = b with b ≠ ±1 associated with this differential equation will have a unique solution. (c) Find all equilibrium solutions to this differential equation.