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mathematics
precalculus
Questions and Answers of
Precalculus
Suppose that a piston is moving straight up and down and that its position at time t sec is s = Acos (2πbt), with A and b positive. The value of A is the amplitude of the motion, and b is the
The position of a particle moving along a coordinate line is s = √1 + 4t, with s in meters and t in seconds. Find the particle’s velocity and acceleration at t = 6 sec.
Suppose that the velocity of a falling body is y = k√s m/sec (k a constant) at the instant the body has fallen s m from its starting point. Show that the body’s acceleration is constant.
The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely
A particle moves along the x-axis with velocity dx/dt = ƒ(x). Show that the particle’s acceleration is ƒ(x)ƒ′(x).
Using the Chain Rule, show that the Power Rule (d/dx)xn = nxn-1 holds for the functions xn. x3/4 = √x√x VxVx
Find dy/dx.y = -10x + 3 cos x
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Find dy/dx.y = x2 cos x
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Find dy/dx.y = 2√x sec x + 3
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. p(0) = √30; p'(1), p'(3), p'(2/3)
Find dy/dx. y csc x 4√x + = - 7 et
Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified. k(z) = 1–²; k'(-1), k'(1), k' (√2) 27
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Find dy/dx. y = x² cotx - 1 x²
Give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.a. Find the body’s displacement and average velocity for the given time interval.b. Find the
Find dy/dx. g(x) = COS X sin² x
Find dy/dx.ƒ(x) = sin x tan x
At time t, the position of a body moving along the s-axis is s = t3 - 6t2 + 9t m.a. Find the body’s acceleration each time the velocity is zero.b. Find the body’s speed each time the acceleration
At time t Ú 0, the velocity of a body moving along the horizontal s-axis is y = t2 - 4t + 3.a. Find the body’s acceleration each time the velocity is zero.b. When is the body moving forward?
Find dy/dx.y = xe-x sec x
The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t2 on Mars and s = 11.44t2 on Jupiter. How long does it take a rock falling from rest to reach
Galileo developed a formula for a body’s velocity during free fall by rolling balls from rest down increasingly steep inclined planks and looking for a limiting formula that would predict a
Find dy/dx.y = (sin x + cos x) sec x
A rock thrown vertically upward from the surface of the moon at a velocity of 24 m / sec (about 86 km / h) reaches a height of s = 24t - 0.8t2 m in t sec.a. Find the rock’s velocity and
Find dy/dx. y 4 COS X + 1 tan x
Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m >sec. Because the acceleration of gravity at the
The accompanying figure shows the velocity y = ds/dt = ƒ(t) (m/ sec) of a body moving along a coordinate line.a. When does the body reverse direction?b. When (approximately) is the body moving at a
Find dy/dx. y = COS X X + X COS X
A particle P moves on the number line shown in part (a) of the accompanying figure. Part (b) shows the position of P as a function of time t.a. When is P moving to the left? Moving to the right?
Find y′ (a) By applying the Product Rule(b) By multiplying the factors to produce a sum of simpler terms to differentiate. y = (x² + 1)(x + 5 + + 1)(x + 5 X
A 45-caliber bullet shot straight up from the surface of the moon would reach a height of s = 832t - 2.6t2 ft after t sec. On Earth, in the absence of air, its height would be s = 832t - 16t2 ft
Find the derivatives of the function. y 2x + 5 3x - 2
Had Galileo dropped a cannonball from the Tower of Pisa, 179 ft above the ground, the ball’s height above the ground t sec into the fall would have been s = 179 - 16t2.a. What would have been the
Find y′ (a) By applying the Product Rule and (b) By multiplying the factors to produce a sum of simpler terms to differentiate.y = (3 - x2) (x3 - x + 1)
When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive
Find the derivatives of the function. g(x) = x² - 4 x + 0.5
The accompanying figure shows the velocity y = ƒ(t) of a particle moving on a horizontal coordinate line.a. When does the particle move forward? Move backward? Speed up? Slow down?b. When is the
Find y′ (a) By applying the Product Rule and (b) By multiplying the factors to produce a sum of simpler terms to differentiate.y = (2x + 3) (5x2 - 4x)
Find the derivatives of the function. Z 4 - 3x 3x² + x
Find dy/dx.y = (sec x + tan x) (sec x - tan x)
The multiflash photograph in the accompanying figure shows two balls falling from rest. The vertical rulers are marked in centimeters. Use the equation s = 490t2 (the freefall equation for s in
Find the values of the derivative. ds dt t=-1 if s 13t² =
Find dy/dx.y = x2 cos x - 2x sin x - 2 cos x
Find the derivatives of the function. f(t) || = 1²2 t² + t - 2 2 1 ▬
Find y′ (a) By applying the Product Rule and (b) By multiplying the factors to produce a sum of simpler terms to differentiate.y = (1 + x2)(x3/4 - x-3)
The graphs in the accompanying figure show the position s, velocity y = ds/dt, and acceleration a = d2s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is
Find dy/dx.ƒ(x) = x3 sin x cos x
Find dy/dx.g(x) = (2 - x) tan2 x
Find ds/dt. S = sin t 1 - cos t
The graphs in the accompanying figure show the position s, the velocity y = ds/dt, and the acceleration a = d2s/dt2 of a body moving along a coordinate line as functions of time t. Which graph is
Find the derivatives of the function. f(s) = Vs - 1 Vs + 1
Find ds/dt.s = tan t - e-t
Find the derivatives of the function. υ บ 1 + x = 4√x X
Suppose that the revenue from selling x washing machines isdollars.a. Find the marginal revenue when 100 machines are produced.b. Use the function r′(x) to estimate the increase in revenue that
Find the derivatives of the function. И 5x + 1 2√x
Find the derivatives of the function.y = (1 - t) (1 + t2)-1
Find ds/dt.s = t2 - sec t + 5et
It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formulaa. Find the rate dy/dt (m/h) at
Find the derivatives of the function. r = 1 2 =(√/ + 0 + Ve
Find the derivatives of the function.w = (2x - 7)-1(x + 5)
Find dr/dθ.r = 4 - θ2 sin θ
Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x - 0.1x2.a. Find the average cost per machine of producing the first 100 washing machines.b. Find the marginal cost
Find dr/dθ.r = θ sin θ + cos θ
Find dp/dq. P = 5 + 1 cot q
Find dr/dθ.r = sec θ csc θ
Find dr/dθ.r = (1 + sec θ) sin θ
A typical male’s body surface area S in square meters is often modeled by the formula S = 1/60√wh, where h is the height in cm, and w the weight in kg, of the person. Find the rate of change of
Find the derivatives of the function. y || 1 (x² - 1) (x² + x + 1)
Give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds/dt = ƒ′(t) and the acceleration function
Find the derivatives of the function. y (x + 1)(x (x - 1)(x + 2) 1)(x − 2)
Find dp/dq.p = (1 + csc q)cos q
Find dp/dq. P sin q + cos q cos q
Find the derivatives of the function. y = x² + 3ex 2ex X
Find dp/dq. P 3q + tan q q sec q
The given graph shows the temperature T in °F at Davis, CA, on April 18, 2008, between 6 a.m. and 6 p.m.a. Estimate the rate of temperature change at the timesi) 7 a.m. ii) 9 a.m. iii) 2 p.m. iv)
The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)2. How fast is the water running out at the end of 10 min? What is the average rate at
The graph in the accompanying figure shows the average annual percentage change y = ƒ(t) in the U.S. gross national product (GNP) for the years 2005–2011. Graph dy/dt (where defined).
Graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation. y = sin x, -3π/2 ≤ x ≤ 2π x = -π, 0, 3π/2
Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v2, where s is measured in ft and v in mph. The
Find the derivatives of the function.y = 2e-x + e3x
The volume V = (4/3)πr3 of a spherical balloon changes with the radius.a. At what rate (ft3/ft) does the volume change with respect to the radius when r = 2 ft?b. By approximately how much does the
Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t2, where D is measured in meters from the starting point and t is measured in seconds from the time
Graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation. y= y = sec x, -π/2
Find the derivatives of the function.y = x3ex
Give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds/dt = ƒ′(t) and the acceleration function
Graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation. y = tan x, x = -π/3, 0, π/3 π/2 < x < π/2
Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single vent in the crater’s
Find the derivatives of the function.w = re-r
Find y″ ifa. y = csc x. b. y = sec x.
Find the derivatives of the function. r = es S
Find the derivatives of the function. = A Vx2 - x X
Find the derivatives of the function.y = x9/4 + e-2x
Graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation. y = 1 + cos x, -3π/2 ≤ x ≤ 2π x = -π/3, 3π/2 X
Find the derivatives of the function. W = 1 1.4 + πT Vz
Find y(4) = d4 y/dx4 ifa. y = -2sin x.b. y = 9cos x.
Give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds/dt = ƒ′(t) and the acceleration function
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