Application of Differentiation Rules (Chapter 4)

Question 1

What is the Extreme Value Theorem?

Answer 1

If f is continunous on a closed interval [a,b] then f has both a minimum and a maximum on the interval.

Question 2

What is the definition of a critical number?

Answer 2

Let f be defined at c. If f’(c) = 0 or if f is not differentiable at c, then c is a critical number of f. Relative extrema occur only at critical numbers.

Question 3

What is the process for finding extrema on a closed interval?

Answer 3

To find the extrema of a continuous function f on a closed interval [a,b] use the following steps: 1. Find the critical numbers of f in (a,b) 2. Evaluate f at each critical number in (a,b) 3. Evaluate f at each endpoint of [a,b] 4. The least of these is the minimum, the greatest is the maximum.

Question 4

What is Rolle’s Theorem?

Answer 4

If f is continuous on the closed interval [a,b} and differentiable on the open interval (a,b): Then, if f(a) = f(b) there is at least one number c in (a,b) such that f’(c) = 0

Question 5

What is the Mean Value Theorem?

Answer 5

If f is continuous on the closed interval [a,b] and fifferentiable on the open interval (a,b) then there exists a number c in (a,b) such that: f’(c) = f(b) - f(a) / b - a

Question 6

What is the alternative form of the Mean Value Theorem?

Answer 6

f(b) = f(a) + (b - a)f’(c)

Question 7

How do you determine if a function is increasing or decreasing?

Answer 7

Use the Test for Increasing and Decreasing functions (Theorem 4.5) Let f be a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). 1. If f’(x) > 0 for all x in (a,b) then f is increasing on [a,b]. 2. If f’(x) < 0 for all x in (a,b) then f is decreasing on [a,b]. 3. If f’(x) = 0 for all x in (a,b) then f is constant on [a,b].

Question 8

What is the table for the first derivative test?

Answer 8

Interval: Test Value: Sign of f’(x): Conclusion:

Question 9

What are the guidelines for finding intervals on which a function is increasing or decreasing?

Answer 9

Let f be continuous on the interval (a,b). To find the open intervals on which f is increasing or decreasing: 1. Locate the critical numbers of f in (a,b) and use these numbers to determine test intervals 2. Determine the sign of f’(x) at one test value in each of the intervals 3. Use Theorem 4.5 to determine whether f is increasing or decreasing on each interval These guidelines are also valid if the interval (a,b) is replaced by an interval of the form (-infinity,b), (a,infinity) or (-infinity, infinity).

Question 10

What is the first derivative test?

Answer 10

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then: 1. If f’(x) changes from negative to positive at c, then f has a relative maximum at (c,f(c)) 2. If f’(x) changes from positive to negative at c, then f has a relative maximum at (c,f(c)) 3. If f’(x) is positive or negative on both sides then f(c) is neither a relative minimum or relative maximum

Question 11

What is the definition of of concavity?

Answer 11

Let f be differentiable on an open interval I. The graph of f is concave upward on I if f’ is increasing on the interval and concave downward on I if f’ is decreasing on the interval.

Question 12

What is the test for concavity (Theorem 4.7)?

Answer 12

Let f be a function whose second derivative exists on an open interval I: 1. If f’‘(x) > 0 for all x in I, then the graph of f is concave upward on I 2. If f’‘(x) < 0 for all x in I, then the graph of f is concave downward on I

Question 13

What is the table for the test for concavity?

Answer 13

Interval: Test Value: Sign of f’‘(x): Conclusion

Question 14

What is a point of inflection?

Answer 14

A point of inflection is a point on the graph of where the concavity changes from upward to downward or downward to upward

Question 15

How do you find points of inflection?

Answer 15

Set the second derivative equal to 0 (or undefined) and solve for x. This *MAY* yield a point of inflection, you must test it to make sure there is a concavity change.

Question 16

What is the second derivative test (Theorem 4.9)?

Answer 16

Let f be a function such that f’(c) = 0 and the second derivative of f exists on an open interval containing c. 1. If f’‘(c) > 0 then f(c) is relative minimum 2. If f’‘(c) < 0 then f(c) is a relative maximum If f’‘(c) = 0, the test fails. If this happens, use the First Derivative Test.

Question 17

What is the definition of a Horizontal Asymptote?

Answer 17

The line y = L is a horizontal asymptote of the graph of f if: Lim x->infinity f(x) = L or Lim x->(neg)infinity f(x) = L

Question 18

What are the limits at infinity (Theorem 4.10)?

Answer 18

If r is a positive rational number and c is any real number, then: 1. Lim x->Infinity c/x^r = 0 and Lim x->(Neg)Infinity c/x^r = 0 The second limit (Negative Infinity) is valid only if x^r is defined when x < 0. 2. Lim x->(Neg)Infinity e^x=0 and Lim x->Infinity e^-x=0

Question 19

What are the guidelines for finding limits at +/-Infinity of Rational Functions?

Answer 19

1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. 2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

Question 20

How do you find Lim x->Infinity sin x / x?

Answer 20

The range is -1 <= 1/x. Both 1/x = 0, so by Squeeze theorem sin x/x = 0.

Question 21

What are the guidelines for analyzing the graph of a function?

Answer 21

1. Determine the domain and range of the function 2. Determine the intercepts, asymptotes and symmetry of the graph 3. Locate the x-values for which f’(x) and f’‘(x) are either zero or do not exist. Use the results to determine relative extrema and points of inflection.

Question 22

What do you need to find to sketch a graph?

Answer 22

First Derivative Second Derivative x-intercepts y-intercepts Vertical Asymptotes Horizontal Asymptotes Critical Numbers Possible Points of Inflection Domain Test Intervals End Behavior

Question 23

What are the guidelines for solving applied minimum and maximum problems?

Answer 23

1. Identify all given quantities and all quantities to be determined. If possible, make a sketch. 2. Write a primary equation for the quantity that is to be maximized or minimized 3. Reduce the primary equation to one having a single independent variable. This may involve the use of secondary equations relating the independent variables of the primary equation. 4. Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense. 5. Determine the desired maximum or minimum value by the calculus techniques discussed in sections 4.1 through 4.4

Question 24

What is the table to use for curve sketching?

Answer 24

Question 25

How can you find Lim x->infinity x sin 1/x?

Answer 25

Let x = 1/t and find the limit as t=>0+

Question 26

How can you find: Lim x->(neg)infinity (x + sqrt(x^2 + 3)?

Answer 26

Treat the expression as a fraction whose denominator is 1 and rationalize the numerator.

Question 27

What is the process for finding two positive numbers with a given product, another equation, and knowing one of them is a minimum?

Answer 27

1. Back substitute one variable into the non-product equation so you are only working with 1 variable
2. Take the derivative of the resulting equation and set it equal to 0 to find a critical number.
3. Take the second derivative of that equation and set it equal to 0 to determine if the number is a minimum or maximum.
4. Plug the resulting value of the variable back into the product equation to determine the other variable

Question 28

What is the formula for a tangent line approximation?

Answer 28

y = f(c) + f’(c)(x-c)

Question 29

How do you find actual change in y?

Answer 29

f(c + delta X) - f(c)

Question 30

How do you find approximate change in y?

Answer 30

delta y (approx) = f’c(delta x) = f’(x) dx

Question 31

How do you find propagated error?

Answer 31

Let delta x = measurement error. Find f(x + delta x) - f(x).

Question 32

How do you find relative error? How do you find percent error?

Answer 32

Take the ratio of the derivative of the desired quantity to the function for the desired quantity (4pir^2 dr / 4/3 pi r^3) and simplify (3 dr / r). Then plug in the variable (r) and the differential (dr) and solve. Percent error is relative error divided by 100.

Question 33

What formula do you use for approximating function values, and what is the key to using this formula?

Answer 33

f(x + delta x) (approx)= f(x) + dy = f(x) + f’(x) dx. They key is to choose a value for x that makes the calculations easier.

Question 34

How can you use differentials to approximate sqrt(16.5)?

Answer 34

Make f(x) = sqrt(x) and use the above equation to get sqrt(x) + 1 / 2sqrt(x) dx. Choose x = 16 and dx = 0.5 and plug in.