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|
 | 1.
The derivative is important because of its many different uses and interpretations. Describe four aspects of
the derivative: graphical (think of tangent lines), symbolic (the derivative function), numerical (approximations) and applications (velocity and others). |
| | 2. Mathematicians often use the word “smooth” to describe functions with certain (desirable) properties. Graphically, how are differentiable functions smoother than functions that are continuous but not differentiable, or functions which are not continuous? |
| | 3. Briefly describe what
the derivative tells you about the original function. In particular, if the
derivative is positive at a point, what do you know about the trend of the function at that point? What is different if
the derivative is negative at the point? |
| | 4. In
Example 2.3, we found that the derivative of f (x) = 3x-5 is f ' (x) = 3. Explain in terms of
slope why this is true. |
| In exercises 5-8, compute f ' (a) using the limits
(2.1) and (2.2).
|
| 5. f (x) = 3x+1, a = 1 | 6. f (x) = 3x2+1, a = 1 |
| 7.  | 8. 
|
| In exercises 9-12, compute f ' (a) using limit
(2.1) or (2.2).
|
| 9. f (x) = x2+2x, a = 0 | 10. f (x) = x2+2x, a = 3 |
| 11. f (x) = x3+4, a = -1 | 12. f (x) = x4-3x, a = -2
|
| In exercises 13-20, compute
the derivative function
f ' (x) using (2.1) or
(2.2). |
| 13. f (x) = 3x2+1 | 14. f (x) = x2-2x+1 |
| 15.  | 16.  |
| 17.  | 18. f (x) = 2x+3 |
| 19. f (x) = x3+2x-1 | 20. f (x) = x4-2x2+1
|
| In exercises 21-26, match the graphs of the functions on the left with the graphs of their derivatives on the right.
|
| 21.
| (a)
|
| 22.
| (b)
|
| 23.
| (c)
|
| 24.
| (d)
|
| 25.
| (e)
|
| 26.
| (f)
|
| In exercises 27-30, use the given graph of
f (x) to sketch a graph of f ' (x).
|
| 27.
| 28.
|
| 29.
| 30.
|
| In exercises 31 and 32, use the given graph of f ' (x) to sketch a plausible graph of
f (x).
|
| 31.
| 32.
|
| In exercises 33 and 34, identify all points at which the
function is not differentiable.
|
| 33.
| 34.
|
| In exercises 35 and 36, estimate the
slope of the tangent line at x = 1.
|
| 35. |
|
| x | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 |
| f(x) | 2.4 | 3.1 | 3.9 | 4.8 | 5.8 | 6.8 | 7.7 | 8.5 | 9.2 |
|
| 36. |
|
| x | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 |
| f(x) | 4.0 | 4.6 | 5.3 | 6.1 | 7.0 | 7.8 | 8.6 | 9.3 | 9.9 |
|
| |
| In exercises 37 and 38, use the distances
f (t ) to estimate the velocity at t = 2.
|
| 37. |
|
| t | 1.6 | 1.7 | 1.8 | 1.9 | 2.0 | 2.1 | 2.2 | 2.3 | 2.4 |
| f(t ) | 2.4 | 3.1 | 3.9 | 4.8 | 5.8 | 6.8 | 7.7 | 8.5 | 9.2 |
|
| 38. |
|
| t | 1.6 | 1.7 | 1.8 | 1.9 | 2.0 | 2.1 | 2.2 | 2.3 | 2.4 |
| f(t ) | 4.0 | 4.6 | 5.3 | 6.1 | 7.0 | 7.8 | 8.6 | 9.3 | 9.9 |
|
| |
| 39. If f (x) = x2/3 , show graphically and numerically that f (x) is continuous at x = 0 but f '(0) does not exist. |
| 40. If  show graphically and numerically that f (x) is continuous at x = 0 but f '(0) does not exist. |
| In exercises 41 and 42, use a CAS or graphing calculator.
|
| 41. Numerically estimate f '(1) for f (x) = xx and verify your answer using a CAS. |
| 42. Numerically estimate f '( ) for f (x) = xsin x and verify your answer using a CAS. |
| In exercises 43 and 44, compute the
right-hand derivative
 and the left-hand derivative 
|
| 43. 
|
| 44. 
|
| 45. Assume that  If f (x) is continuous at x = 0 and g(x) and k(x) are differentiable at x = 0 , prove that D+f (0) = k'(0) and D-f (0) = g'(0). Which statement is not true if f (x) has a jump discontinuity at x = 0 ? |
| 46. Explain why
the derivative f '(0) exists if and only if the one-sided derivatives exist and are equal. |
| 47. If f ' (x) > 0 for all x, use the tangent line interpretation to argue that f (x) is an increasing function; that is, if a < b, then f (a) < f (b).
|
| 48. If f ' (x) < 0 for all x, use the tangent line interpretation to argue that f (x) is a decreasing function; that is, if a < b, then f (a) > f (b).
|
| 49. One model for the spread of a disease assumes that at first the disease spreads very slowly, gradually the infection rate increases to a maximum, then the infection rate decreases back to zero, marking the end of the epidemic. If I(t ) represents the number of people infected at time t, sketch a graph of both I(t ) and I' (t ) , assuming that those who get infected do not recover. |
| 50. One model for urban population growth assumes that at first, the population is growing very rapidly, then the growth rate decreases until the population starts decreasing. If P(t ) is the population at time t, sketch a graph of both P(t ) and P' (t ). |
| 51. Give an example showing that the following is not true for all functions f: if f (x)  x , then f ' (x) 1. |
| 52. Determine if the following is true for all functions f: if f (0) = 0 , f ' (x) exists for all x and f (x) x , then f ' (x) 1. |
| In exercises 53-56, give the units for
the derivative function.
|
| 53. f (t ) represents position, measured in meters, at time t seconds. |
| 54. c(t ) represents the amount of a chemical present, in grams, at time t minutes. |
| 55. f (x) represents the demand, in number of items, of a product when the price is x dollars. |
| 56. p(x) represents the mass, in kg, of the first x meters of a pipe. |
| 57. Psychologists study the way humans think by analyzing the results of experiments like the following. If you take the shape  and rotate it, does it rotate into the shape  or the shape  ? To which shape does  rotate? Research indicates that the length of time to answer such questions is proportional to the angle the given figure needs to rotate to reach a target figure. The conclusion is that we actually mentally rotate the given figure. (Did you do this consciously?) Pinker measures the speed of rotation as 11° per microsecond (0.001 second). Given this rate, how long should it take to correctly identify  ?  ? Did you find  easier to identify? |
 | 58. Compute
the derivative function for x2, x3 and x4. Based on your results, identify the pattern and conjecture a general formula for
the derivative of xn. Test your conjecture on the functions  = x1/2 and 1/x = x-1. |
| | 59. In
Theorem 2.1, it is stated that a differentiable
function is guaranteed to be continuous. The converse is nottrue: continuous functions are not necessarily
differentiable (see Example 2.9). This fact is carried to an extreme in Weierstrass' function, to be explored here. First, graph the
function f 4(x) = cos x +  cos 3x +  cos 9x +  cos 27x +  cos in the graphing window 0 x 2 and -2 y 2. Note that the graph appears to have several sharp corners, where
a derivative would not exist. Next, graph the function f 6(x) = f 4(x)+ cos 243x+ cos 729x.
Note that there are even more places where the graph appears to have sharp corners. Explore graphs of f 10(x), f 14(x) and so on, with more terms added. Try to give graphical support to the fact that the Weierstrass
function f  (x) is continuous for all x but is not differentiable for any x. More graphical evidence comes from the
fractal nature of the Weierstrass function: compare the graphs of f 4(x) with 0 x 2 and -2 y 2 and f 6(x) - cos x- cos 3x with  and - y . Explain why the graphs are identical. Then explain why this indicates that no matter how much you zoom in on a graph of the Weierstrass
function, you will continue to see wiggles and corners. That is, you cannot zoom in to find a tangent line.
|
| | 60. Suppose there is a
function F(x) such that F(1) = 1 and F(0) = f 0, where 0 < f 0 < 1. If F'(1) > 1 , show graphically that the equation F(x) = x has a solution q where 0 < q < 1. (Hint: Graph y = x and a plausible F(x) and look for intersections.) Sketch a graph where F'(1) < 1 and there are no solutions to the equation F(x) = x between 0 and 1 (although x = 1 is a solution). Solutions have a connection with the probability of the extinction of animals or family names. Suppose you and your descendants have children according to the following probabilities: f 0 = 0.2 is the probability of having no children, f 1 = 0.3 is the probability of having exactly one child, and f 2 = 0.5 is the probability of having two children. Define F(x) = 0.2+0.3x+0.5x2 and show that F'(1) > 1. Find the solution of F(x) = x between x = 0 and x = 1 ; this number is the probability that your “line” will go extinct some time into the future. Find nonzero values of f 0 , f 1 and f 2 such that the corresponding F(x) satisfies F'(1) < 1 and hence the probability of your line going extinct is 1. |
