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mathematics
calculus

Questions and Answers of Calculus

An object thrown directly downward from the top of a cliff with an initial velocity of v0 feet per second falls s = v0t + 16t2 feet in t seconds. If it strikes the ocean below in 3 seconds with a
An object moves along a horizontal coordinate line in such a way that its position at time t is specified by s = t3 - 3t2 - 24t - 6. Here s is measured in centimeters and t in seconds. When is the
Explain why an object moving along a line is slowing down when its velocity and acceleration have opposite signs (see Problem 37).
Use the formula of Problem 39 to find D4x(x4 sin x). In problem Leibniz obtained a general formula for Dnx(uv), where u and v are both function of x. See if you can find it
Let f(x) = x[sin x - cos (x/2)]. (a) Draw the graphs of f(x), f'(x), f"(x), and f"'(x) on [0, 6] using the same axes. (b) Evaluate f"'(2.13).
Repeat Problem 41 for f(x) = (x + 1) / (x2 + 2). (a) Draw the graphs of f(x), f'(x), f"(x), and f"'(x) on [0, 6] using the same axes. (b) Evaluate f"'(2.13).
In problems 1-3, find f"(2). 1. f(x) = x2 + 1 2. f(x) = 5x3 + 2x2 + x 3. f(t) = 2/t
Assuming that each equation in Problems 1-3 defines a differentiable function of x, find Dxy by implicit differentiation. 1. y2 - x2 = 1 2. 9x2 + 4y2 = 36 3. xy = 1
In problem 1-3, find the equation of the tangent line at the indicated point (see Example 3). 1. x3y + y3x = 30; (1, 3) 2. x2y2 + 4xy = 12y; (2, 1)
In Problems 1-3, find dy/dx. 1. y = 3x5/3 + √x 2. y = 3√x - 2x7/2 3. y = 3√x + 1/ 3√x 4. y = 4√2x + 1
If s2t + t3 = 1, find ds/dt and dt/ds.
Sketch the graph of the circle x2 + 4x + y2 + 3 = 0 and then find equations of the two tangent line that pass through the origin.
Find the equation of the normal line (line perpendicular to the tangent line) to the curve 8(x2 + y2)2 = 100(x2 - y2) at (3, 1).
Suppose that xy + y3 = 2. Then implicit differentiation twice with respect to x yields in turn: (a) xy' + y + 3y2y' = 0' (b) xy" + y' + y' + 3y2y" + 6y(y')2 = 0.
Find y" if x3 - 4y2 + 3 = 0 (see Problem 37).
Fin y" at (2, 1) if 2x2y - 4y3 = 4 (see Problem 37).
Use implicit differentiation twice to find y" at (3, 4) if x2 + y2 = 25.
Show that the normal line to x3 + y3 = 3xy at (3/2, 3/2) passes through the origin.
Show that the hyperbolas xy = 1 and x2 - y2 = 1 intersect at right angles.
Show that the graph of 2x2 + y2 = 6 and y2 = 4x intersect at right angles.
Suppose that curves C1 and C2 intersect at (x0, y0) with slopes m1 and m2, respectively, as in Figure 4. Then (see Problem 40 of Section 0.7) the positive angle Î¸ from C1 (i.e., from the
Find the angle from the line y = 2x to the curve x2 - xy + 2y2 = 28 at their point of intersection in the first quadrant (see Problem 44).
A particle of mass m moves along the x-axis so that its position x and velocity v = dx/dt satisfy m(v2 - v20) = k(x20 - x2) where v0, x0, and k are constants. Show by implicit differentiation that m
The curve x2 - xt + y2 = 16 is an ellipse centered at the origin and with the line y = x as its major axis. Find the equations of the tangent lines at the two points where the ellipse intersects the
Find all points on the curve x2y - xy2 = 2 where the tangent line is vertical, that is, where dx/dy = 0.
How high h must the light bulb in Figure 5 be if the point (1, 25, 0) is on the edge of the illuminated region?
Each edge of a variable cube is increasing at a rate of 3 inches per second. How fast is the volume of the cube increasing when an edge is 12 inches long?
A child is flying a kite. If te kite is 90 feet above the child's hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the child paying out cord when 150
A rectangular swimming pool is 40 feet long, 20 feet wide, 8 feet deep at the deep end, and 3 feet deep at the shallow end (see Figure 10). If the pool is filled by pumping water into it at the rate
A particle P is moving along the graph of y = √x2 - 4, x ≥ 2, so that the x-coordinate of P is increasing at the rate of 5 units per second. How fast is the y-coordinate of P increasing when x =
A metal disk expands during heating. If its radius increases at the rate of 0.02 inch per second, how fast is the area of one of its faces increasing when its radius is 8.1 inches?
Two ships sail from the same island port, one going north at 24 knots (24 nautical miles per hour) and the other east 30 knots. The northbound ship departed at 9:00 A.M. and the east bound ship left
A light in a lighthouse 1 kilometers offshore from a straight shoreline is rotating at 2 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point ½ kilometer
An aircraft spotter observes a plane flying at a constant altitude of 4000 feet toward a point directly above her head. She notes that when the angle of elevation is f/2 radian it is increasing at a
Chris, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 feet high at a rate of 2 feet per second. (a) How fast is his shadow increasing in length when Chris is
The vertex angle θ opposite the base of an isosceles triangle with equal sides of length 100 centimeters is increasing at 1/10 radian per minute. How fast is the area of the triangle increasing when
A long, level highway bridge passes over a railroad track that is 100 feet below it and at right angles to it. If an automobile traveling 45 miles per hour (66 feet per second) is directly above a
Assuming that a soap bubble retains its spherical shape as it expands, how fast is its radius increasing when its radius is 3 inches if air is blown into it at a rate 3 cubic inches per second?
Water is pumped at a uniform rate of 2 liters (1 liter = 1000 cubic centimeters) per minute into a tank shaped like a frustum of a right circular cone. The tank has altitude 80 centimeters and lower
Water is leaking out the bottom of a hemispherical tank of radius 8 feet at a rate of 2 cubic feet per hour. The tank was full at a certain time. Flow fast is the water level changing when its height
The hands on a clock are of length 5 inches (minute hand) and 4 inches (hour hand). How fast is the distance between the tips of the hands changing at 3:00?
A steel ball will drop 16t2 feet in t seconds. Such a ball is dropped from a height of 64 feet at a horizontal distance 10 feet from a 48-foot street light. How fast is the ball's shadow moving when
Rework Example 6 assuming that the water tank is a sphere of radius 20 feet. (See Problem 21 for the volume of a spherical segment.)
Rework Example 6 assuming that the water tank is in the shape of an upper hemisphere of radius 20 feet. (See Problem 21 for the volume of a spherical segment.)
Refer to Example 6. How much water did Webster City use during this 12-hour period from midnight to noon? Hint: This is not a differentiation problem.
An 18-foot ladder leans against a 12-foot vertical wall, its top extending over the wall. The bottom end of the ladder is pulled along the ground away from the wall at 2 feet per second. (a) Find the
A spherical steel ball rests at the bottom of the tank of Problem 21. Answer the question posed there if the ball has radius (a) 6 inches (b) 2 feet. (Assume that the ball does not affect the flow
A snowball melts at a rate proportional to its surface area. (a) Show that its radius shrinks at a constant rate. (b) If it melts to 4 its original volume in one hour, how long will it take to melt
An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer
A right circular cylinder with a piston at one end is filled lied with gas. Its volume is continually changing because of the movement of the piston If the temperature of the gas is kept constant,
A girl 5 feet tall walks toward a street light 20 feet high at a rate of 4 feet per second. Her little brother, 3 feet tall, follows at a constant distance of 4 feet directly behind her (Figure
A student is using a straw to drink from a conical paper cup, whose axis is vertical, at a rate of 3 cubic centimeters per second. If the height of the cup is 10 centimeters and the diameter of its
An airplane flying west at 300 miles per hour goes over the control tower at noon, and a second airplane at the same altitude, flying north at 400 miles per hour, goes over the tower an hour later.
A woman on a dock is pulling in a rope fastened to the bow of a small boast. If the women's hands are 10 feet higher than the point where the rope is attached to the boat and if she is retrieving the
A 20-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the level pavement directly away from the building at 1 foot per second, how fast is the top of the ladder
We assume that an oil spill is being cleaned up by deploying bacteria that consume the oil at 4 cubic feet per hour. The oil spill itself is modeled in the form of a very thin cylinder whose height
Sand is pouring form a pipe at the rate of 16 cubic feet per second. If the falling sand forms a conical pile on the ground whose altitude is always ¼ the diameter of the base, how fast is the
In problem 1-8, find dy. 1. y = x2 + x - 3 2. y = 7x3 + 3x2 + 1 3. y = (2x + 3)-4 4. y = (3x2 + x + 1)-2 5. y = (sin x + cos x)3 6. y = (tan x + 1)3
For the function defined in Problem 1D, make a careful drawing of the graph off for - 1.5 ≤ x ≤ 15 and the tangents to the curve at x = 0.5 and x = - 1; on this drawing label dy and dx for each
Let y = 1/x. Find the value of dy in each case. (a) x = 1, dx = 0.5 (b) x = - 2, dx = 0.75
For the function defined in Problem 12, make a careful drawing (as in Problem 11) for - 3 ≤ .x < 0 and 0 < x ≤ 3.
If y = x2 - 3, find the values of Δy and dy in each case. (a) x = 2 and dx = Δx = 0.5 (b) x = 3 and dx = Δx = - 0.12
If y = x4 + 2.1, find the values of Δy and dy in each case. (a) x = 2 and dx = Δx = 1 (b) x = 2 and dx = Δx = 0.005
In Problems 1-2, use differentials to approximate the given number (see Example 2). Compare with calculator values. 1. √402 2. √35.2
The outside diameter of a thin spherical shell is 12 feet. If the shell is 03 inch thick, use differentials to approximate the volume of the region interior to the shell.
The interior of an open cylindrical tank is 12 feet in diameter and 8 feet deep. The bottom is copper and the sides are steel. Use differentials to find approximately how many gallons of
The period of a simple pendulum of length L feet is given by T = 2π √L/g seconds. We assume that g, the acceleration due to gravity on (or very near) the surface of the earth, is 32 feet per
The diameter of a sphere is measured as 20 ± 0.1 centimeters. Calculate the volume and estimate the absolute error and the relative error (see Example 4).
A cylindrical roller is exactly 12 inches long and its diameter is measured as 6 ± 0.005 inches. Calculate its volume with an estimate for the absolute error and the relative error.
The angle θ between the two equal sides of an isosceles triangle measures 0.53 ± 0.005 radian. The two equal sides are exactly 151 centimeters long. Calculate the length of the third side with an
Calculate the area of the triangle of Problem 29 with an estimate for the absolute error and the relative error. A = 1 / 2ab sin θ.
It can be shown that if |d2y / dx2| ≤ M on a closed interval with c and c + Ax as end points, then |Δy - dy| ≤ 1/2M(Δx)2 Find, using differentials, the change in y = 3x2 - 2x + 11 when x
Suppose that f is a function satisfying f(1) = 10, and f'(1.02) = 12. Use this information to approximate 1(1.02).
Suppose f is a function satisfying f(3) = 8 and f'(3.05) = 1/4. Use this information to approximate f(3.05).
A conical cup, 10 centimeters high and 8 centimeters wide at the top, is filled with water to a depth of 9 centimeters. An ice cube 3 centimeters on a side is about to be dropped in. Use
A tank has the shape of a cylinder with hemispherical ends. If the cylindrical part is 100 centimeters long and has an out-side diameter of 20 centimeters, about how much paint is required to coat
Einstein's Special Theory of Relativity says that an object's mass in is related to its velocity v by the formulaHere m0 is the rest mass and c is the speed of light. Use differentials to determine
In Problems 1-2, find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. 1. f(x) = x2 at a = 2, [0,
Find the linear approximation to f(x) = mx + b at an arbitrary et. What is the relationship between f(x) and L(x)?
Show that for every a > 0 the linear approximation L(x) to the function f(x) = x2 at a satisfies f(x) ≤ L(x) for all 0.
Show that for every a the linear approximation L(x) to the function f = x2 at a satisfies L(x) ≤ f(x) for all x.
Find a linear approximation to f(x) = (1 + x)α at x = 0. where a is any number. For various values of a, plot f(x) and its linear approximation L(x). For what values of a does the linear
Suppose f is differentiable. If we use the approximation f(x + h) f(x) f'(x) h the error is e(h) = f(x h) - f(x) - f'(x) h. Show that
In Problems 1-4, find all critical points and find the minimum and maximum of the function. Each function has domain [-2, 4].1.2. 3. 4.
Identify the critical points and find the extreme values on the interval [-1, 5] for each function: (a) f(x) = x3 - 6x2 + x + 2 (b) g(x) = |f(x)|
Identify the critical points and find the extreme values on the interval [- 1, 5] for each function: (a) f(x) = cos x + x sin x + 2 (b) g(x) = |f(x)|
f is differentiable, has domain [0. 6], reaches a maximum of 6 (attained when x = 3) and a minimum of 0 (attained when x = 0). Additionally, x = 5 is a stationary point. Sketch the graph of a
f is differentiable, has domain [0, 6], reaches a maximum of 4 (attained when x = 6) and a minimum of - 2 (attained when x - 1). Additionally, x = 2, 3, 4, 5 are stationary points. Sketch the graph
f is continuous, but not necessarily differentiable, has domain [0, 6], reaches a maximum of 6 (attained when x = 5), and a minimum of 2 (attained when x = 3). Additionally, x = 1 and x = 5 are the
f is continuous, but not necessarily differentiable, has domain [0. 6], reaches a maximum of 4 (attained when x = 4), and a minimum of 2 (attained when x - 2). Additionally, f has no stationary
f is differentiable, has domain [0, 6], reaches a maximum of 4 (attained at two different values of x, neither of which is an end point), and a minimum of 1 (attained at three different values of x,
f is continuous but not necessarily differentiable, has do-main [0, 6], reaches a maximum of 6 (attained when x = 0) and a minimum of 0 (attained when x = 6). Additionally, f has two stationary
F has domain [0, 6] but is not necessarily continuous, and f does not attain a maximum.
F ahs domain [0, 6] but is not necessarily continuous, and f attains neither a maximum nor a minimum.
In Problems, 1-2, identify the critical points and find the maximum value and minimum value on the given interval. 1. f(x) = x2 + 4x + 4; I = [-4, 0] 2. h(x) = x2 + x; I = [-2, 2]
In Problems 1-3, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. 1. f(x) = 3x + 3 2. g(x) = (x + 1) (x - 2) 3. h(t) = t2 + 2t - 3
In Problems 1-3, use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. 1. f(x) = (x - 1)2 2. G(w) = w2 - 1 3.
In Problems 1-2, determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). 1. f(x) = x3 - 12x + 1 2. g(x) = 4x3 -
In Problems 1-2, sketch the graph of a continuous function f on [0, 6] that satisfies all the stated conditions. 1. F(0) = 1; f(6) = 3; increasing and concave down on (0, 6) 2. F(0) = 8; f(6) = - 2;
Prove that a quadratic function has no point of inflection.

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