Chapter 4(differentiation)
-
Upload
eko-wijayanto -
Category
Education
-
view
402 -
download
2
description
Transcript of Chapter 4(differentiation)
BMM 104: ENGINEERING MATHEMATICS I Page 1 of 17
CHAPTER 4: DIFFERENTIATION
The Derivative as a Function
Derivative Function
The derivative of the function f(x) with respect to the variable x is the function whose value at x is
provided the limit exists.
Example: Attend lecture.
PROBLEM SET: CHAPTER 4
Find the following indicated derivatives by using definition.
1. 4.
2. 5.
3. 6.
ANSWERS FOR PROBLEM SET: CHAPTER 4
BMM 104: ENGINEERING MATHEMATICS I Page 2 of 17
Find the following indicated derivatives by using definition.
1. 4.
2. 5.
3. 6.
Differentiation Rules
Derivative of a Constant Function
If f has the constant value f(x) = c, then
Power Rule for Positive Integers
If n is a positive integer, then
Constant Multiple Rule
BMM 104: ENGINEERING MATHEMATICS I Page 3 of 17
If u is a differentiable function of x, and c is a constant, then
Derivative Sum Rule
If u and v are differentiable functions of x, then their sum u + v is differentiable at every point
where u and v are both differentiable. At such points,
Derivative Product Rule
BMM 104: ENGINEERING MATHEMATICS I Page 4 of 17
If u and v are differentiable at x, then so is their product uv, and
Derivative Quotient Rule
If u and v are differentiable at x and if , then the quotient u/v is differentiable at x, and
Example: Attend lecture.
PROBLEM SET: CHAPTER 4
Derivative Calculations
Find the first and second derivatives for the following functions.
1. 7.
2. 8.
3. 9.4. 10.
5. 11.
6. 12.
In the following questions, find (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate.
1. 3.
2. 4.
Differentiate the following functions.
1. 7.
BMM 104: ENGINEERING MATHEMATICS I Page 5 of 17
2. 8.
3. 9.
4. 10.
5. 11.
6. 12.
ANSWERS FOR PROBLEM SET: CHAPTER 4
Derivative Calculations
1. -2 7.2. 2 8.3. 9.4. 10.5. 11.6. 12.
1. 3.2. 4.
1. 7.
2. 8.
3. 9.
4. 10.
5. 11.
6. 12.
BMM 104: ENGINEERING MATHEMATICS I Page 6 of 17
Derivatives of Trigonometric Functions
1.
2.
3.
4.
5.
6.
Example: Attend lecture.
PROBLEM SET 3.3
PROBLEM SET: CHAPTER 4
Find .
1. 7.
2. 8.
3. 9.
4. 10.
5. 11.6. 12.
ANSWERS FOR PROBLEM SET: CHAPTER 4
1. 7.
2. 8.
BMM 104: ENGINEERING MATHEMATICS I Page 7 of 17
3. 9.
4. 10.
5. 11.6. 12.
The Chain Rule
If f(u) is differentiable at the point u = g(x) and g(x) is differentiable at x, then the composite function is differentiable at x, and
In Leibniz’s notation, if y = f(u) and u = g(x), then
where dy/du is evaluated at u = g(x).
Example: Attend lecture.
PROBLEM SET: CHAPTER 4
Differentiate the following functions.
1. 21.2. 22.
3. 23.
4. 24.
5. 25.
6. 26.
7. 27.
8. 28.
BMM 104: ENGINEERING MATHEMATICS I Page 8 of 17
9. 29.
10. 30.
11. 31.12. 32.
13. 33.
14. 34.
15. 35.
16. 36.
17. 37.
18. 38.
19. 39.
20. 40.
ANSWERS FOR PROBLEM SET: CHAPTER 4
1. 21.
2. 22.
3. 23.
4. 24.
5. 25.
6. 26.
BMM 104: ENGINEERING MATHEMATICS I Page 9 of 17
7. 27.
8. 28.
9. 29.
10. 30.
11. 31.
12. 32.
13. 33.
14. 34.
15. 35.
16. 36.
17.
18.
19. 37.
38.
39.
20. 40.
The Derivatives of
BMM 104: ENGINEERING MATHEMATICS I Page 10 of 17
Generally, if u is a differentiable function of x whose values are positive, so that ln u is defined, then applying the Chain Rule
to the function gives
,
Example: Attend lecture.
PROBLEM SET: CHAPTER 4
Find the derivative of y with respect to x, t, or , as appropriate for the following functions.
1. 16.
2. 17.
3. 18.
4. 19.
5. 20.
6. 21.
7. 22.
8. 23.
9. 24.
BMM 104: ENGINEERING MATHEMATICS I Page 11 of 17
10. 25.
11. 26.12. 27.
13. 28.
14. 29.
15. 30.
ANSWERS FOR PROBLEM SET: CHAPTER 4
1. 16.
2. 17.
3. 18.
4. 19.
5. 20.
6. 21.
7. 22.
8. 23.
9. 24.
10. 25.
11. 26.
12. 27.
13. 28.
14. 29.
BMM 104: ENGINEERING MATHEMATICS I Page 12 of 17
15. 30.
The Derivative of
Example: Attend lecture.
PROBLEM SET: CHAPTER 4
Find the derivative of y with respect to x, t, or , as appropriate for the following functions.
1.
2.
3.4.5.6.7.8.9.10.
11.12.13.14.
15.
16.
17.18.
BMM 104: ENGINEERING MATHEMATICS I Page 13 of 17
Find .
1. 2.3. 4.
ANSWERS FOR PROBLEM SET: CHAPTER 4
1. 10.
2. 11.
3. 12.
4. 13.
5. 14.
6. 15.
7. 16.
8. 17.
9. 18.
Find .
1. 3.
2. 4.
Monotonic Functions, the First Derivative and Second Derivative Test for Concavity and CurveSketching
Definitions: Increasing, Decreasing Function
BMM 104: ENGINEERING MATHEMATICS I Page 14 of 17
Let f be a function defined on an interval I and let and be any two points in I.
1. If whenever < , then f is said to be increasing on I.2. If whenever < , then f is said to be decreasing on I.
A function that is increasing or decreasing on I is called monotonic on I.
First Derivative Test for Monotonic Functions
Suppose that f is continuous on [a, b] and differentiable on (a, b).
If at each point , then f is increasing on [a, b].If at each point , then f is decreasing on [a, b].
First Derivative Test for Local Extrema
Suppose that c is a critical point of a continuous function f, and that f is differentiable at every point in some interval containing c expect possibly at c itself. Moving across c from left to right.
1. if changes from negative to positive at c, then f has a local minimum at c;
2. if changes from positive to negative at c, then f has a local maximum at c;
3. if does not change sign at c (that is, is positive on both sides of c or negative on both sides), then f has no local extremum at c.
BMM 104: ENGINEERING MATHEMATICS I Page 15 of 17
The Second Derivative Test for Concavity
Let y = f(x) be twice-differentiable on an interval I.1. If on I, the graph of f over I is concave up.2. If on I, the graph of f over I is concave down.
Definition: Point of Inflection
A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.
Second Derivative Test for Local Extrema
Inflection point
BMM 104: ENGINEERING MATHEMATICS I Page 16 of 17
Suppose is continuous on an open interval that contains x = c.1. If and , then f has a local maximum at x = c.2. If and , then f has a local minimum at x = c.3. If and , then the test fails. The function f may have a local
maximum, a local minimum, or neither.
Graph Sketching
Strategy for Graphing y = f(x)1. Identify the domain of f and any symmetries the curve may have.2. Find and .3. Find the critical points of f, and identify the function’s behavior at each one.4. Find where the curve is increasing and where it is decreasing.5. Find the points of inflection, if any occur, and determine the concavity of the
curve.6. Identify any asymptotes.7. Plot key points, such as the intercepts and the points found in Steps 3-5, and
sketch the curve.
Example: Attend lecture
PROBLEM SET: CHAPTER 4
Sketch the graph for the following functions.
1.2.3.
4.
BMM 104: ENGINEERING MATHEMATICS I Page 17 of 17
5.
ANSWERS FOR PROBLEM SET: CHAPTER 4
Solution: Attend lecture.