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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Show that for any real number k, (1 + Δx)k ≈ 1 + kΔx for small Δx. Estimate (1.02)0.7 and (1.02)−0.3.
Let Δƒ= ƒ(1 + h) − ƒ(1), where ƒ(x) = x−1. Show directly that E = |Δƒ − ƒ'(1)h| is equal to h2/(1 + h). Then prove that E ≤ 2h2 if −1/2 ≤ h ≤ 1/2. In this case, 1/2 ≤ 1 + h
In Exercises 9–12, refer to Figure 13.Estimate ƒ(1) andƒ'(2). 3.0 2.5 2.0 1.5 1.0 0.5 y 0.5 1.0 f(x). 1.5 2.0 2.5 3.0 FIGURE 13 ·x
Compute dy/dx using the limit definition.y = 1/2 − x
Refer to a 5-m ladder sliding down a wall, as in Figures 5 and 6. The variable h is the height of the ladder’s top at time t, and x is the distance from the wall to the ladder’s bottom.Figure
Calculate the derivative with respect to x of the other variable appearing in the equation.3y3 + x2 = 5
Let ƒ(x) = (2x2 − 5)2. Compute ƒ'(x) three different ways: 1. Multiplying out and then differentiating, 2. using the Product Rule, and 3. using the Chain Rule. Show that the results coincide.
Compute the derivative.ƒ(θ) = tan θ sec θ
Calculate y" and y"'.y = θ2(2θ + 7)
In Exercises 9–12, refer to Figure 13.Find a value of h for which 3.0 2.5 2.0 1.5 1.0 0.5 y 0.5 1.0 f(x). 1.5 2.0 2.5 3.0 FIGURE 13 ·x
Compute dy/dx using the limit definition.y = 1/(x − 1)2
Calculate the derivative with respect to x of the other variable appearing in the equation.y4 − 2y = 4x3 + x
Let ƒ(x) = (x + sin x)−1. Compute ƒ'(x) separately using the Quotient Rule and the Chain Rule. Show that the results coincide.
Compute the derivative.k(θ) = θ2 sin2 θ
Express the limit as a derivative. lim h→0 √1+h-1 h
Suppose that the top is sliding down the wall at a rate of 1.2 m/s. Calculate dx/dt when h = 3 m.
In Exercises 13–16, refer to Figure 14.Determine ƒ'(a) for a = 1, 2, 4, 7. 4 3 2. 1 2 4 5 6 7 8 9 FIGURE 14 Graph of f. 1 3 X
Suppose that h(0) = 4 and the top slides down the wall at a rate of 1.2 m/s. Calculate x and dx/dt at t = 2 s.
Calculate the derivative with respect to x of the other variable appearing in the equation.x2y + 2x3y = x + y
Compute the derivative using derivative rules that have been introduced so far.y = (x4 + 5)3
Compute the derivative.ƒ(x) = (2x4 − 4x−1) sec x
Express the limit as a derivative. x³ + 1 X² 3 lim x1 x + 1
Calculate y" and y"'.y = x − 4/x
In Exercises 13–16, refer to Figure 14.For which values of x is ƒ'(x) 4 3 2. 1 2 4 5 6 7 8 9 FIGURE 14 Graph of f. 1 3 X
What is the relation between h and x at the moment when the top and bottom of the ladder move at the same speed?
Compute the derivative using derivative rules that have been introduced so far.y = (8x4 + 5)3
Calculate the derivative with respect to x of the other variable appearing in the equation.xy2 + x2y5 − x3 = 3
Compute the derivative.ƒ(z) = z tan z
Express the limit as a derivative. lim 1- sint cost t-π
Calculate y" and y"'.y = 1/1 − x
In Exercises 15–18, compute ƒ'(x) and find an equation of the tangent line to the graph at x = a.ƒ(x) = x4, a = 2
In Exercises 13–16, refer to Figure 14.Which is larger, ƒ'(5.5) or ƒ'(6.5)? 4 3 2. 1 2 4 5 6 7 8 9 FIGURE 14 Graph of f. 1 3 X
The radius r and height h of a circular cone change at a rate of 2 cm/s. How fast is the volume of the cone increasing when r = 10 and h = 20?
Compute the derivative. y = sec 0 0
Compute the derivative using derivative rules that have been introduced so far.y = √7x − 3
Calculate the derivative with respect to x of the other variable appearing in the equation.x3R5 = 1
Express the limit as a derivative. lim 0→ cos sin 0 +1 0-л
Calculate y" and y"'.y = s−1/2(s + 1)
A road perpendicular to a highway leads to a farmhouse located 2 km away (Figure 9). An automobile travels past the farmhouse at a speed of 80 km/h. How fast is the distance between the automobile
In Exercises 15–18, compute ƒ'(x) and find an equation of the tangent line to the graph at x = a.ƒ(x) = x−2, a = 5
In Exercises 13–16, refer to Figure 14.Show that ƒ'(3) does not exist. 4 3 2. 1 2 4 5 6 7 8 9 FIGURE 14 Graph of f. 1 3 X
Compute the derivative. G(z) = 1 tanz - cotz
Compute the derivative using derivative rules that have been introduced so far.y = (4 − 2x − 3x2)5
Calculate the derivative with respect to x of the other variable appearing in the equation.x4 + z4 = 1
Calculate y" and y"'.y = (r1/2 + r)(1 − r)
Find ƒ(4) and ƒ'(4) if the tangent line to the graph of ƒ at x = 4 has equation y = 3x − 14.
A man of height 1.8 m walks away from a 5-m lamppost at a speed of 1.2 m/s (Figure 10). Find the rate at which his shadow is increasing in length. 5 X У
In Exercises 15–18, compute ƒ'(x) and find an equation of the tangent line to the graph at x = a.ƒ(x) = 5x − 32√x, a = 4
Sketch a graph of velocity as a function of time for the shuttle train in Example 6.Example 6 EXAMPLE 6 Describe the motion and velocities of a shuttle train that runs on a straight track at the
In Exercises 17–20, use the limit definition to calculate the derivative of the linear function.ƒ(x) = 7x − 9
Compute the derivative. R(y) = 3 cos y - 4 sin y
Compute the derivative using derivative rules that have been introduced so far.y = (x2 + 9x)−2
Calculate the derivative with respect to x of the other variable appearing in the equation. y X + = У : 2y
Each graph in Figure 2 shows the graph of a function ƒ and its derivative ƒ'. Determine which is the function and which is the derivative. (1) A B ^ (II) FIGURE 2 Graph of f. (III) X
Calculate the derivative indicated.g"'(−1), g(t) = −4t−5
As Claudia walks away from a 264-cm lamppost, the tip of her shadow moves twice as fast as she does. What is Claudia’s height?
Compute the derivative using derivative rules that have been introduced so far.y = (2 cos θ + 5 sinθ)9
Compute the derivative. f(x) = sin x sin x + 1 1
Calculate the derivative with respect to x of the other variable appearing in the equation.sin(xt) = t
A laser pointer is placed on a platform that rotates at a rate of 20 revolutions per minute. The beam hits a wall 8 m away, producing a dot of light that moves horizontally along the wall. Let θ be
Let N(t) be the percentage of a state population infected with a flu virus on week t of an epidemic. What percentage is likely to be infected in week 4 if N(3) = 8 and N'(3) = 1.2?
Find ƒ(3) and ƒ'(3), assuming that the tangent line to y = ƒ(x) at a = 3 has equation y = 5x + 2.
Compute the derivative using derivative rules that have been introduced so far.y = √9 + x + sin x
Compute the derivative. f(x) = csc² t t
In Exercises 15–18, compute ƒ'(x) and find an equation of the tangent line to the graph at x = a.ƒ(x) = 3√x, a = 8
In Exercises 17–20, use the limit definition to calculate the derivative of the linear function.ƒ(x) = 12
Compute the derivative using derivative rules that have been introduced so far.y = (x3 + 3x + 9)−4/3
Compute the derivative. f(x) = X sin x + 2
Calculate the derivative with respect to x of the other variable appearing in the equation. √x + s 1 X + 1 S
In the setting of Exercise 19, let θ be the angle that the line through the radar station and the plane makes with the horizontal. How fast is θ changing 12 min after the plane passes over the
Compute the derivative using derivative rules that have been introduced so far.y = cos(9θ + 41)
Compute the derivative.ƒ(θ) = θ tan θ sec θ
Calculate the derivative with respect to x of the other variable appearing in the equation. у+ У 12 + x
Sketch the graph of a continuous function ƒ if the graph of ƒ' appears as in Figure 5 and ƒ(0) = 0. 2 2 + 3 + 4 →X
A hot air balloon rising vertically is tracked by an observer located 4 km from the lift-off point. At a certain moment, the angle between the observer’s line of sight and the horizontal is π/5,
Is (A), (B), or (C) the graph of the derivative of the function ƒ shown in Figure 3? them (A) A (B) y = f(x) 2 X LAV (C) -x
Fred X has to make a book delivery from his warehouse, 15 mi north of the city, to the Amazing Book Store 10 mi south of the city. Traffic is usually congested within 5 mi of the city. He leaves at
At a given moment, a plane passes directly above a radar station at an altitude of 6 km.(a) The plane’s speed is 800 km/h. How fast is the distance between the plane and the station changing half a
Calculate the derivative with respect to x of the other variable appearing in the equation.x1/2 + y2/3 = −4y
Sketch the graph of ƒ' if the graph of ƒ appears as in Figure 4. 2- 1 | 2 3 نیا 4
Find a point on the graph of y = √x where the tangent line has slope 10.
In Exercises 17–20, use the limit definition to calculate the derivative of the linear function.g(t) = 8 − 3t
Calculate the derivative with respect to x of the other variable appearing in the equation.y−2/3 + x3/2 = 1
Compute the derivative using derivative rules that have been introduced so far.y = cos4 θ
Compute the derivative. f(x) = 1 + tan x 1 - tan x
At the start of the 27th century, the population of Zosania was approximately 40 million. Early-century prosperity saw the population nearly double in the first three decades, but the growth slowed
In Exercises 17–20, use the limit definition to calculate the derivative of the linear function.k(z) = 14z + 12
Find an equation of the tangent line at x = 3, assuming that ƒ(3) = 5 and ƒ'(3) = 2.
Calculate the derivative indicated.ƒ'''(4), ƒ(t) = 2t2 − t
Calculate the derivative with respect to x of the other variable appearing in the equation.sin(x + y) = x + cos y
Calculate the derivative with respect to x of the other variable appearing in the equation.tan(x2y) = (x + y)3
Compute the derivative. R(0) = cos 4 + cos 0
Compute the derivative using derivative rules that have been introduced so far.y = sin (√x2 + 2x + 9)
Describe the tangent line at an arbitrary point on the graph of y = 2x + 8.
A girl’s height h(t) (in centimeters) is measured at time t (in years) for 0 ≤ t ≤ 14:(a) What is the average growth rate over the 14-yr period?(b) Is the average growth rate larger over the
A rocket travels vertically at a speed of 1200 km/h. The rocket is tracked through a telescope by an observer located 16 km from the launching pad. Find the rate at which the angle between the
Suppose that ƒ(2 + h) − ƒ(2) = 3h2 + 5h. Calculate:(a) The slope of the secant line through (2, ƒ (2)) and (6, ƒ (6))(b) ƒ'(2)
Compute the derivative using derivative rules that have been introduced so far.y = tan(4 − 3x) sec(3 − 4x)
Using a telescope, you track a rocket that was launched 4 km away, recording the angle θ between the telescope and the ground at half-second intervals. Estimate the velocity of the rocket if θ(10)
Compute the derivative. g(z) = cot z 3-3 sin z

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