The acceleration due to gravity, g, varies with height above the surface of the earth

For the graphs in Exercises 12, list the x-values for which the function appears to be

For the graphs in Exercises 12, list the x-values for which the function appears to be

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

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

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.

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.

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.

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? 20 10 0 10 20

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

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 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 reason

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?

A function f that is not differentiable at x = 0 has a graph with a sharp corner at x = 0.

If f is not differentiable at a point then it is not continuous at that point.

A continuous function that is not differentiable at x = 2

An invertible function that is not differentiable at x = 0.

A rational function that has zeros at x = 1 and is not differentiable at x = 2.

There is a function which is continuous on [1, 5] but not differentiable at x = 3.

If a function is differentiable, then it is continuous.

If a function is continuous, then it is differentiable.

If a function is not continuous, then it is not differentiable

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