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study help
mathematics
precalculus
Questions and Answers of
Precalculus
What do the limits limh→0 ((sin h)/h) and limh→0 ((cos h - 1)/h) have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?
Find the derivatives of the function.r = √2θ sin θ
Let x and y be differentiable functions of t and let s = √x2 + y2 be the distance between the points (x, 0) and (0, y) in the xy-plane.a. How is ds/dt related to dx/dt if y is constant?b. How is
Find the linearization of ƒ(x) = √x + 1 + sin x at x = 0. How is it related to the individual linearizations of √x + 1 and sin x at x = 0?
Find the derivatives of the function. r = sin (0+ V0 + 1) /0
Find the derivatives of the function. y = 2√x sin √x
Once you know the derivatives of sin x and cos x, how can you find the derivatives of tan x, cot x, sec x, and csc x? What are the derivatives of these functions?
Find the derivatives of the function.r = √2θcos θ
If x, y, and z are lengths of the edges of a rectangular box, the common length of the box’s diagonals is s = √x2 + y2 + z2.a. Assuming that x, y, and z are differentiable functions of t, how is
Is there anything special about the derivative of an odd differentiable function of x? Give reasons for your answer.
Find the derivatives of the function y 1 2 ===x²³ csc ²7/ CSC X
At what points are the six basic trigonometric functions continuous? How do you know?
Find the derivatives of the function.r = sin √2θ
Is there anything special about the derivative of an even differentiable function of x? Give reasons for your answer.
What is the rule for calculating the derivative of a composite of two differentiable functions? How is such a derivative evaluated? Give examples.
When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm / min. At what rate is the plate’s area increasing when the radius is 50 cm?
Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates:Find the rates at which the box’s (a) Volume, (b) Surface area(c) Diagonal length s =
Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0, that ƒ is differentiable at x0, that ƒ(x0) = 0, and that g is continuous at x0. Show that the
Is the derivative ofcontinuous at x = 0? How about the derivative of k(x) = xh (x)? Give reasons for your answers. h(x) = [x² sin (1/x), x0 10, x = 0
A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.a. How fast is the top of the
If u is a differentiable function of x, how do you find (d/dx)(un) if n is an integer? If n is a real number? Give examples.
The length ∫ of a rectangle is decreasing at the rate of 2 cm/sec while the width w is increasing at the rate of 2 cm/sec. When ∫ = 12 cm and w = 5 cm, find the rates of change of (a) The
What is implicit differentiation? When do you need it? Give examples.
Use mathematical induction to prove that if y = u1u2 · · · un is a finite product of differentiable functions, then y is differentiable on their common domain and dy dx = du₁ dx -U₂・・ Un
What is the derivative of the natural logarithm function ln x? How does the domain of the derivative compare with the domain of the function?
Find the derivatives of the functiony = x-1/2 sec (2x)2
What is the derivative of the exponential function ax, a > 0 and a ≠ 1? What is the geometric significance of the limit of (ah - 1)/h as h → 0? What is the limit when a is the number e?
Find the derivatives of the function.y = √x csc (x + 1)3
Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots
A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft away from her?
What is the derivative of loga x? Are there any restrictions on a?
Find the derivatives of the function.y = 5 cot x2
The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-thousandth of an inch every 3 min. How
Leibniz’s rule for higher-order derivatives of products of differentiable functions says thata.b.c.The equations in parts (a) and (b) are special cases of the equation in part (c). Derive the
Which of the expressions are defined, and which are not? Give reasons for your answers.a. sec-1 0 b. sin-1√2
Find the derivatives of the function y = 4x√x + √x
In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a
Find the derivatives of the function. r sin 0 Cos 0 - - 1 2
Find the derivatives of the function. r = 1 + sin 0 1 - cos 0 2
You are videotaping a race from a stand 132 ft from the track, following a car that is moving at 180 mi/h (264 ft/sec), as shown in the accompanying figure. How fast will your camera angle θ be
Find the derivatives of the function. y = (2x + 1)√2x + 1
A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. How fast is the shadow of the ball moving along the ground 1/2 sec
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
Find the derivatives of the function y 3 (5x²+ sin 2x)³/2
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
The coordinates of a particle in the metric xy-plane are differentiable functions of time t with dx/dt = -1 m/sec and dy/dt = -5 m/sec. How fast is the particle’s distance from the origin changing
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
Find the derivatives of the functiony = 20 (3x - 4)1/4 (3x - 4)-1/5
You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is 12 ft long and starts 3 ft from the wall you are sitting next to. Show that your
On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 ft long. At the moment in question, the angle u the sun makes with the ground is
Find the derivatives of the functiony = (3 + cos3 3x)-1/3
A baseball diamond is a square 90 ft on a side. A player runs from first base to second at a rate of 16 ft/sec.a. At what rate is the player’s distance from third base changing when the player is
A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in3/min, how fast is the thickness of the ice decreasing when it is 2
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
a. Here is a pictorial proof that sec-1 (-x) = π - sec-1 x. See if you can tell what is going on.b. Derive the identity sec-1 (-x) = π - sec-1 x by combining the following two equations from the
Find the derivatives of the function y 1 xetx 4 1 16 e4x
Find the derivatives of the function.y = 10e-x/5
Each function ƒ(x) changes value when x changes from x0 to x0 + dx. Finda. The change Δƒ = ƒ(x0 + dx) - ƒ(x0);b. The value of the estimate dƒ = ƒ′(x0) dx; andc. The approximation error |Δƒ
A highway patrol plane flies 3 mi above a level, straight road at a steady 120 mi/h. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane
Here is an informal proof that tan-1 1 + tan-1 2 + tan-1 3 = π. Explain what is going on.
Find the derivatives of the function.y = √2e√2x
Two ships are steaming straight away from a point O along routes that make a 120° angle. Ship A moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yd). Ship B moves at 21 knots. How
Find the derivatives of the function.y = x2e-2/x
At what rate is the angle between a clock’s minute and hour hands changing at 4 o’clock in the afternoon?
Write a differential formula that estimates the given change in volume or surface area.The change in the volume V = (4/3)πr3 of a sphere when the radius changes from r0 to r0 + dr
Find the derivatives of the function.y = ln (sin2 θ)
An explosion at an oil rig located in gulf waters causes an elliptical oil slick to spread on the surface from the rig. The slick is a constant 9 in. thick. After several days, when the major axis of
Write a differential formula that estimates the given change in volume or surface area.The change in the volume V = x3 of a cube when the edge lengths change from x0 to x0 + dx
Find the derivatives of the function.y = ln (sec2 θ)
Write a differential formula that estimates the given change in volume or surface area.The change in the surface area S = 6x2 of a cube when the edge lengths change from x0 to x0 + dx
Which of the expressions are defined, and which are not? Give reasons for your answers.a. tan-1 2 b. cos-1 2
Find the derivatives of the function.y = log2 (x2/2)
Which of the expressions are defined, and which are not? Give reasons for your answers.a. csc-1 (1/2) b. csc-1 2
Write a differential formula that estimates the given change in volume or surface area.The change in the lateral surface area S = πr√r2 + h2 of a right circular cone when the radius changes from
Find the derivatives of the function.y = log5 (3x - 7)
Use the Derivative Rule in Section 3.8, Theorem 3, to deriveSection 3.8, Theorem 3 d dx sec-¹x = 1 |x|√x² - 1' |x > 1.
Write a differential formula that estimates the given change in volume or surface area.The change in the volume V = πr2h of a right circular cylinder when the radius changes from r0 to r0 + dr and
Find the derivatives of the functiony = 8-t
Which of the expressions are defined, and which are not? Give reasons for your answers.a. cot-1 (-1/2) b. cos-1 (-5)
Write a differential formula that estimates the given change in volume or surface area.The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h0 to
Estimate the volume of material in a cylindrical shell with length 30 in., radius 6 in., and shell thickness 0.5 in. 6 in. K -30 in. 0.5 in.
Find the derivatives of the function.y = 92t
The radius of a circle is increased from 2.00 to 2.02 m.a. Estimate the resulting change in area.b. Express the estimate as a percentage of the circle’s original area.
Find the derivatives of the function.y = 5x3.6
The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree’s diameter increase? The tree’s cross-sectional area?
Find the derivatives of the function.y = √2x-√2
Find the derivatives of the function.y = (x + 2)x+2
A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in
The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, dθ/dt = -0.6 rad/sec.a. How fast is the light moving along the shore when it reaches
Points A and B move along the x- and y-axes, respectively, in such a way that the distance r (meters) along the perpendicular from the origin to the line AB remains constant. How fast is OA changing,
Find the derivatives of the function. y = sin ¹V1 u², 0
What is special about the functionsExplain. f(x) = sin-¹ x-1 x + 1' x ≥ 0, and g(x) = 2 tan¹ √x?
What is special about the functionsExplain. 1 √x² + 1 f(x) = sin-1. and g(x) tan-¹ = ܕܥܗ
Use the formula in Exercise 55 to find dy/dx ifa. y = (sin-1 x)2b.Exercise 55Assume that y = sin-1 x is a differentiable function of x. By differentiating the equation x = sin y implicitly, show that
Find the derivatives of the function.y = 2(ln x)x/2
Find the derivatives of the function. y = sin 1 Vu v > 1
Assume that y = sin-1 x is a differentiable function of x. By differentiating the equation x = sin y implicitly, show that dy/dx = 1/√1 - x2.
The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’sa. Circumference? b. Area?
Find the derivatives of the function -1 y = z cos¹z-VI-z² Z Z
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