A magnetic field, B, is given as a function of the distance,r, from the center of a wire

For the graphs in Exercises 12, list the x-values for which the function appears to be (a) Not continuous. (b) Not differentiable. 1

For the graphs in Exercises 12, list the x-values for which the function appears to be (a) Not continuous. (b) Not differentiable. 2

In Exercises 34, does the function appear to be differentiable on the interval of x-values shown? 3

In Exercises 34, does the function appear to be differentiable on the interval of x-values shown? 4

Decide if the functions in Problems 57 are differentiable at x = 0. Try zooming in on a graphing calculator, or calculating the derivative f (0) from the definition. f(x)=(x + |x|) 2 + 1

Decide if the functions in Problems 57 are differentiable at x = 0. Try zooming in on a graphing calculator, or calculating the derivative f (0) from the definition. f(x) =  x sin(1/x) + x for x = 0 0 for x = 0

Decide if the functions in Problems 57 are differentiable at x = 0. Try zooming in on a graphing calculator, or calculating the derivative f (0) from the definition. f(x) =  x2 sin(1/x) for x = 0 0 for x = 0

In each of the following cases, sketch the graph of a continuous function f(x) with the given properties. (a) f(x) > 0 for x < 2 and for x > 2 and f (2) is undefined. (b) f(x) > 0 for x < 2 and f(x) < 0 for x > 2 and f (2) is undefined.

Look at the graph of f(x)=(x2 + 0.0001)1/2 shown in Figure 2.59. The graph of f appears to have a sharp corner at x = 0. Do you think f has a derivative at x = 0?

The acceleration due to gravity, g, varies with height above the surface of the earth, in a certain way. If you go down below the surface of the earth, g varies in a different way. It can be shown that g is given by g = GMr R3 for r<R GM r2 for r R where R is the radius of the earth, M is the mass of the earth, G is the gravitational constant, and r is the distance to the center of the earth. (a) Sketch a graph of g against r. (b) Is g a continuous function of r? Explain your answer. (c) Is g a differentiable function of r? Explain your answer.

An electric charge, Q, in a circuit is given as a function of time, t, by Q =  C for t 0 Cet/RC for t > 0, where C and R are positive constants. The electric current, I, is the rate of change of charge, so I = dQ dt . (a) Is the charge, Q, a continuous function of time? (b) Do you think the current, I, is defined for all times, t? [Hint: To graph this function, take, for example, C = 1 and R = 1.]

A magnetic field, B, is given as a function of the distance, r, from the center of a wire as follows: B = r r0 B0 for r r0 r0 r B0 for r>r0. (a) Sketch a graph of B against r. What is the meaning of the constant B0? (b) Is B continuous at r = r0? Give reasons. (c) Is B differentiable at r = r0? Give reasons.

A cable is made of an insulating material in the shape of a long, thin cylinder of radius r0. It has electric charge distributed evenly throughout it. The electric field, E, at a distance r from the center of the cable is given by E =  kr for r r0 k r2 0 r for r>r0. (a) Is E continuous at r = r0? (b) Is E differentiable at r = r0? (c) Sketch a graph of E as a function of r.

Graph the function defined by g(r) = 1 + cos (r/2) for 2 r 2 0 for r < 2 or r > 2. (a) Is g continuous at r = 2? Explain your answer. (b) Do you think g is differentiable at r = 2? Explain your answer.

The potential, , of a charge distribution at a point on the y-axis is given by = 2 y2 + a2 y  for y 0 2 y2 + a2 + y  for y < 0 where and a are positive constants. [Hint: To graph this function, take, for example, 2 = 1 and a = 1.] (a) Is continuous at y = 0? (b) Do you think is differentiable at y = 0?

Sometimes, odd behavior can be hidden beneath the surface of a rather normal-looking function. Consider the following function: f(x) =  0 if x < 0 x2 if x 0. (a) Sketch a graph of this function. Does it have any vertical segments or corners? Is it differentiable everywhere? If so, sketch the derivative f of this function. [Hint: You may want to use the result of Example 4 on page 94.] (b) Is the derivative function, f (x), differentiable everywhere? If not, at what point(s) is it not differentiable? Draw the second derivative of f(x) wherever it exists. Is the second derivative function, f(x), differentiable? Continuous?

In Problems 1718, explain what is wrong with the statement. A function f that is not differentiable at x = 0 has a graph with a sharp corner at x = 0.

In Problems 1718, explain what is wrong with the statement. If f is not differentiable at a point then it is not continuous at that point.

In Problems 1921, give an example of: A continuous function that is not differentiable at x = 2.

In Problems 1921, give an example of: An invertible function that is not differentiable at x = 0.

In Problems 1921, give an example of: A rational function that has zeros at x = 1 and is not differentiable at x = 2.

Are the statements in Problems 2226 true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. There is a function which is continuous on [1, 5] but not differentiable at x = 3.

Are the statements in Problems 2226 true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is differentiable, then it is continuous

Are the statements in Problems 2226 true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is continuous, then it is differentiable.

Are the statements in Problems 2226 true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is not continuous, then it is not differentiable.

Are the statements in Problems 2226 true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is not differentiable, then it is not continuous

Which of the following would be a counterexample to the statement: If f is differentiable at x = a then f is continuous at x = a? (a) A function which is not differentiable at x = a but is continuous at x = a. (b) A function which is not continuous at x = a but is differentiable at x = a. (c) A function which is both continuous and differentiable at x = a. (d) A function which is neither continuous nor differentiable at x = a.