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Prove that if f is differentiable at a and f(a) 6= 0, then |f| is also differentiable at a. Give an example to show why the assumption f(a) 6= 0 is necessary.
Show that if f is continuous on the closed interval [a, b], then there is a function g which is continuous on R, and which satisfies g(x) = f(x) for all x in [a, b].
A wire loop of radius 0.50 m lies so that an external magnetic field of magnitude 0.30 T is perpendicular to the loop. The field reverses its direction, and its magnitude changes to 0.20 T in 2.0 s.
A 10.7g object moving to the right at 18.3 cm/s makes an elastic head-on collision with a 16.2g object moving in the opposite direction at 28.6 cm/s. What is the velocity of the 16.2g object after t
A mountaineer leaves home at 7am and walks to the top of the mountain, arriving at 7pm. Use the IVT to show that there is a point on the path that he will cross at exactly the same time of day on bo
The transformer supplying power to an artist's light sculpture provides 9600-V AC when supplied by 120-V AC. If there are 100 turns in the transformer's primary coil, how many turns are there in its
A light bulb radiates an electromagnetic wave in all directions with a total power of 100 Watts. If you stand 5 meters away from the bulb, what is the amplitude of the magnetic field (Bmax) of the r
Z is countable, because you can list all the integers in the order 0, 1,-1, 2,-2, 3,-3, ···. Is the set Q of all rational numbers countable? How about R and C?
Two point charges attract each other with an electric force of magnitude F. If one charge is reduced to one-third its original value and the distance between the charges is doubled, what is the resu
Two lightbulbs are in series, one operating at 120 W and the other operating at 60.0 W. If the voltage drop across the series combination is 120 V, what is the current in the circuit?
A series combination of a resistor and a capacitor are connected to a 120-V rms, 69.2-Hz AC source. If the capacitance is 0.896 ?F and the rms current in the circuit is 27.9 mA, what is the resistan
The collision between a hammer and a nail can be considered to be approximately elastic. Calculate the kinetic energy acquired by a 13.9-g nail when it is struck by a 554-g hammer moving with an ini
Sketch and describe the following sets and their images under the function z → z2. The set of all points of the form z = x + 2xi.
The flywheel of a car engine is a solid disk with a mass of 8.7 kg and a radius of 22 cm and is rotating at 4000 rpm. If the entire kinetic energy of the flywheel is converted to gravitational poten
Let r be a real constant greater than 2. The set {z ? C| |z + 1| + |z -1| = r} is a curve in the plane. Describe it and then find its equation in terms of x and y.
A concave spherical mirror has a radius of curvature of 4.0 meters. A real object, 5.0 cm high, is placed 3.0 meters in front of the mirror. By construction and computation determine the position an
If the complex number z is imagined as a point in the complex plane. What are the complex numbers obtained from z by the following geometric transformations?
What is the speed of the water leaving the end of the hose and what is the speed if it were traveling through a nozzle 1/3 its size?
A proton with an initial speed of 800000 m/s is brought to rest by an electric field. A) What was the potential difference that stopped the proton?
Using your answers to the previous part, find the following, expressing your answers first in polar then Cartesian (standard) form.
Find all solutions of the equation z2+ 3 + 4i = 0 by setting z = a + bi for some real numbers a and b. Solve z2+ z + 1 + i = 0.
The diameter of a car tire is 40cm and the diameter of a toy car tire is 18mm. Find the ratio of the diameter of the car to the diameter of the toy car.
Marvin currently has a balance of $1,192 in an account he has held for 14 years. He opened the account with an initial deposit of $800. What is the simple interest on the account?
In a physics lab experiment, a compressed spring launches a 26 g metal ball at a 25 degree angle. Compressing the spring 18 cm causes the ball to hit the floor 1.6 m below the point at which it leav
A charged particle A exerts a force of 2.28 ?N to the right on charged particle B when the particles are 12.1 mm apart. Particle B moves straight away from A to make the distance between them 17.5 m