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mathematics
precalculus
Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions
Find the derivatives of the function.y = x2 cot 5x
Find the derivatives of the function S 4t t + 1 -2
Sand falls from a conveyor belt at the rate of 10 m3/min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) Height and (b) Radius changing when the pile is 4 m high? Answer in centimeters per minute.
How can you write any real power of x as a power of e? Are there any restrictions on x? How does this lead to the Power Rule for differentiating arbitrary real powers?
Find the derivatives of the function.y = x2 sin2 (2x2)
Water is flowing at the rate of 50 m3/min from a shallow concrete conical reservoir of base radius 45 m and height 6 m.a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep?b. How fast is the radius of the water’s surface changing then? Answer in centimeters
What is one way of expressing the special number e as a limit? What is an approximate numerical value of e correct to 7 decimal places?
Find the derivatives of the function y Vx 1 + x 2
Find the derivatives of the function.y = x-2 sin2 (x3)
Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3/min.a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep?b. How fast is the level in the cone falling then? 6" 6" 6" How fast is this level falling? How fast is this level
A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.a. How fast is the boat approaching the dock when 10 ft of rope are out?b. At what rate is the angle θ changing at this instant (see the figure)?
What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions?
Find the derivatives of the function y 21 2√x 2 راه 2√x + 1
A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?
Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop’s radius increases at a constant rate.
Find the derivatives of the function y = x² + x R
How do related rates problems arise? Give examples.
A spherical balloon is inflated with helium at the rate of 100π ft3/min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing?
What is the linearization L(x) of a function ƒ(x) at a point x = a? What is required of ƒ at a for the linearization to exist? How are linearizations used? Give examples.
A particle moves along the parabola y = x2 in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m/sec. How fast is the angle of inclination θ of the line joining the particle to the origin changing when x = 3 m?
If x moves from a to a nearby value a + dx, how do you estimate the corresponding change in the value of a differentiable function ƒ(x)? How do you estimate the relative change? The percentage change? Give an example.
Find the value of ds/du at u = 2 if s = t2 + 5t and t = (u2 + 2u)1/3.
Find the derivatives of the functiony = (x + 1)2(x2 + 2x)
Find the linearizations of the following functions at x = 0.a. sin x b. cos x c. tan x d. ex e. ln (1 + x)
A bus will hold 60 people. The number x of people per trip who use the bus is related to the fare charged (p dollars) by the law p = [3 - (x/40)]2. Write an expression for the total revenue r(x) per trip received by the bus company. What number of people per trip will make the marginal revenue
Describe geometrically when a function typically does not have a derivative at a point.
If x = y3 - y and dy/dt = 5, then what is dx/dt when y = 2?
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = x2 + 2x, a = 0.1
How is a function’s differentiability at a point related to its continuity there, if at all?
Find the derivatives of the function.y = (θ2 + sec θ + 1)3
Find dy/dx. 3 y= y = x + 5 sin x X
Fnd the linearization L(x) of ƒ(x) at x = a.ƒ(x) = x3 - 2x + 3, a = 2
An equation like sin2 θ + cos2 θ = 1 is called an identity because it holds for all values of θ. An equation like sin θ = 0.5 is not an identity because it holds only for selected values of θ, not all. If you differentiate both sides of a trigonometric identity in θ with respect to θ, the
What is the derivative of a function ƒ? How is its domain related to the domain of ƒ? Give examples.
Suppose that the radius r and area A = πr2 of a circle are differentiable functions of t. Write an equation that relates dA/dt to dr/dt.
Find the derivatives of the functiony = x5 - 0.125x2 + 0.25x
Fnd the linearization L(x) of ƒ(x) at x = a.ƒ(x) = √x2 + 9, a = -4
Find the derivatives of the function y = x² + √7x- 1 π + 1
If the identity sin (x + a) = sin x cos a + cos x sin a is differentiated with respect to x, is the resulting equation also an identity? Does this principle apply to the equation x2 - 2x - 8 = 0? Explain.
What role does the derivative play in defining slopes, tangents, and rates of change?
Suppose that the radius r and surface area S = 4πr2 of a sphere are differentiable functions of t. Write an equation that relates dS/dt to dr/dt.
Find the derivatives of the functiony = 3 - 0.7x3 + 0.3x7
Fnd the linearization L(x) of ƒ(x) at x = a.ƒ(x) = x + 1/x, a = 1
a. Find values for the constants a, b, and c that will make ƒ(x) = cos x and g(x) = a + bx + cx2 satisfy the conditions ƒ(0) = g(0), ƒ′(0) = g′(0), and ƒ″(0) = g″(0).b. Find values for b and c that will make ƒ(x) = sin (x + a) and g(x) = b sin x + c cos x satisfy the conditions ƒ(0) =
How can you sometimes graph the derivative of a function when all you have is a table of the function’s values?
Assume that y = 5x and dx/dt = 2. Find dy/dt.
Find the derivatives of the functiony = x3 - 3(x2 + π2)
Fnd the linearization L(x) of ƒ(x) at x = a.ƒ(x) = 3√x, a = -8
The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (-12, -9) and
a. Show that y = sin x, y = cos x, and y = a cos x + b sin x (a and b constants) all satisfy the equation y″ + y = 0.b. How would you modify the functions in part (a) to satisfy the equation y″ + 4y = 0?Generalize this result.
What does it mean for a function to be differentiable on an open interval? On a closed interval?
Find the derivatives of the function y - 1 csc 0 2 82 2 4
Assume that 2x + 3y = 12 and dy/dt = -2. Find dx/dt.
Fnd the linearization L(x) of ƒ(x) at x = a.ƒ(x) = tan x, a = π
Find the derivatives of the function S = Vi 1 + Vi
Find the value. ec (cos-¹1) COS sec
How are derivatives and one-sided derivatives related?
If y = x2 and dx/dt = 3, then what is dy/dt when x = -1?
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = x-1, a = 0.9
What rules do you know for calculating derivatives? Give some examples.
Find the derivatives of the function S = 1 Vt - 1
If x2y3 = 4/27 and dy/dt = 1/2, then what is dx/dt when x = 2?
Find the value. tan sin
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = 2x2 + 3x - 3, a = -0.9
On August 5, 1988, Mike McCarthy of London jumped from the top of the Tower of Pisa. He then opened his parachute in what he said was a world record lowlevel parachute jump of 179 ft. Make a rough sketch to show the shape of the graph of his speed during the jump.
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate. f(x) = X x + 1' a = 1.3
If L = √x2 + y2, dx/dt = -1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = 1 + x, a = 8.1
Find the value. cot sin (-³) 2
If r + s2 + y3 = 12, dr/dt = 4, and ds/dt = -3, find dy/dt when r = 3 and s = 1.
Find dy/dx. COS X y 1 + sin x
particle of constant mass m moves along the x-axis. Its velocity v and position x satisfy the equationwhere k, v0, and x0 are constants. Show that whenever v ≠ 0, ½ m (v² - v₁²) = 1⁄2 k (x² − x²),
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = 3√x, a = 8.5
What is a second derivative? A third derivative? How many derivatives do the functions you know have? Give examples.
Find the derivatives of the function y 1 sin²x ~/. 2 sin x
On Earth, you can easily shoot a paper clip 64 ft straight up into the air with a rubber band. In t sec after firing, the paper clip is s = 64t - 16t2 ft above your hand.a. How long does it take the paper clip to reach its maximum height? With what velocity does it leave your hand?b. On the moon,
If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’sa. Surface area change?b. Volume change?
Find the derivatives of the functiony = 2tan2 x - sec2 x
Find the derivatives of the function S = cot³
Find all values of the constants m and b for which the functionisa. Continuous at x = p.b. Differentiable at x = p. y = sin x, x с п mx + b₂ x ≥ π
What is the derivative of the exponential function ex? How does the domain of the derivative compare with the domain of the function?
At time t sec, the positions of two particles on a coordinate line are s1 = 3t3 - 12t2 + 18t + 5 m and s2 = -t3 + 9t2 - 12t m. When do the particles have the same velocities?
A cube’s surface area increases at the rate of 72 in2/sec. At what rate is the cube’s volume changing when the edge length is x = 3 in?
The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related by the equation V = IR. Suppose that V is increasing at the rate of 1 volt/sec while I is decreasing at the rate of 1/3 amp/sec. Let t denote time in seconds.a. What is the
Does the functionhave a derivative at x = 0? Explain. f(x) = 0, - cos X x x = 0 x = 0
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = e-x, a = -0.1
What is the relationship between a function’s average and instantaneous rates of change? Give an example.
The radius r and height h of a right circular cylinder are related to the cylinder’s volume V by the formula V = πr2h.a. How is dV/dt related to dh/dt if r is constant?b. How is dV/dt related to dr/dt if h is constant?c. How is dV/dt related to dr/dt and dh/dt if neither r nor h is constant?
Find the derivatives of the functions = cos4 (1 - 2t)
Find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.ƒ(x) = sin-1 x, a = π/12
a. Show that if the position x of a moving point is given by a quadratic function of t, x = At2 + Bt + C, then the average velocity over any time interval [t1, t2] is equal to the instantaneous velocity at the midpoint of the time interval.b. What is the geometric significance of the result in part
How do derivatives arise in the study of motion? What can you learn about an object’s motion along a line by examining the derivatives of the object’s position function? Give examples.
The radius r and height h of a right circular cone are related to the cone’s volume V by the equation V = (1/3)πr2h.a. How is dV/dt related to dh/dt if r is constant?b. How is dV/dt related to dr/dt if h is constant?c. How is dV/dt related to dr/dt and dh/dt if neither r nor h is constant?
a. For what values of a and b willbe differentiable for all values of x?b. Discuss the geometry of the resulting graph of ƒ. f(x) = Jax, x < 2 ax² bx + 3, x ≥ 2 -
Show that the linearization of ƒ(x) = (1 + x)k at x = 0 is L(x) = 1 + kx.
How can derivatives arise in economics?
Find the derivatives of the function.s = (sec t + tan t)5
a. For what values of a and b willbe differentiable for all values of x?b. Discuss the geometry of the resulting graph of g. g(x) fax + b₂ x = -1 lax³ + x + 2b, x>-1
The power P (watts) of an electric circuit is related to the circuit’s resistance R (ohms) and current I (amperes) by the equation P = RI2.a. How are dP/dt, dR/dt, and dI/dt related if none of P, R, and I are constant?b. How is dR/dt related to dI/dt if P is constant?
Find the derivatives of the function.s = csc5(1 - t + 3t2)
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