Sketch the graph of a function that is continuous except for the stated discontinuity. Discontinuous, but continuous from the right at 2.

Find the x-value at which f is discontinuous and determine whether f is continuous from the right, or from the left, or neither.

Optimization. What is the Maximum Vertical Distance between y=x+2 and y=x^2 for x in [-1, 2].

The Mean Value Theorem.

Let f(x) = -3x^2+2x+4. Find the value(s) of x that satisfy the Mean Value Theorem on the interval [ -1, 1].

Graph the Secant Line, the Tangent Line, and f(x).

Differentiate y = x ^(SQR x).

Limit involving infinity

Integrals of Even and Odd Functions and the Function Average.

0:00 Integrals of Even and Odd Functions and the Function Average.

0:25 Introduction to Even Functions.

3:56 Introduction to Odd Functions.

6:31 Trig Functions as Odd and Even Functions.

7:58 Neither Odd nor Even?

9:13 Integrals of EVEN Functions RULE.

10:21 Integrals of ODD Functions RULE.

14:05 Example: Integral of Cos (x ) from -pi/2 to pi/2.

16:56 The Average Value of a Function Rule.

22:01 Example: Integral of x^2 from -2 to 2.

23:02 Example: Integral of x^3 from -2 to 2.

23:57 Example: Integral of Cos (x ) from -2Pi to 2Pi.

25:33 Example: Integral of x^9 from -5 to 5.

26:20 Example: Integral of x^2 times Sin (x ) from -Pi to Pi.

27:40 Example: Integral of x^9 -3x^5 + 2x^2 - 10 from -2 to 2.

29:30 Example: Find the Average Value of the Function f(x) = x - x^2 from 0 to 1.

Riemann Sums and Approximating Area under Curves.

0:00 Approximating Area under Curves.

0:10 Riemann Sums.

0:14 Example: The velocity of an object is v = 2t+1.... Find the Displacement.

0:30 Left- Riemann Sums.

7:08 Right- Riemann Sums.

11:48 Midpoint- Riemann Sums.

13:00 Area under the Curve Exact.

14:53 Bonus. 20:00 Example: F(x) = x^2 with n =4. Using table of values and L.R.Sum and R.R.Sum.

25:33 Sigma Notation.

The Mean Value Theorem "MVT".

0:00 The Mean Value Theorem.

0:35 Rolle’s Theorem.

3:23 The Mean Value Theorem "MVT".

5:23 Consequences of the Mean Value Theorem.

5:28 Zero Derivative Implies Constant Function.

9:09 Functions with equal derivatives differ by a constant.

11:36 Example: f(x) = 2 SQRT(x).

14:01 Example: f(x) = x (x-1)^2 over [0,1].

16:36 Example: f(x) = 1-ABS (x) over [-1, 1].

17:23 Example: f(x) = Cos (4x) over [ pi/8, 3pi/ 8].

18:47 Example: f (x) = 1-x^(2/3) over [-1, 1].

19:23 Example: f(x) = x^3-x^2-5x-3 over [-1,3].

20:53 Example: f(x) = 7-x^2 over [-1, 2].

23:45 Example: f(x) = -2x for x less than or equal to 0, and x for x greater than or equal to 0.

25:21 Example: f(x) = ln (2x) over [1, e].

The Chain Rule.

0:00 The Chain Rule.

2:29 Example one in three different ways.

9:39 Example two in detail.

Derivatives of Trigonometric Functions.

0:00 Derivatives of Trigonometric Functions.

0:30 Lim as x approaches 0 of (Sin(x)) / (x).

14:27 Lim as x approaches 0 of (Cos(x)-1) / (x).

17:49 Lim as x approaches 0 of (Sin(3x)) / (x).

19:25 Lim as x approaches 0 of (Sin(3x)) / (Sin(4x)).

20:45 Theorem and proof of d/dx (Sin(x) = Cos (x).

24:10 Proof of d/dx (Tan(x) = Sec^2 (x)).

25:52 Derivatives of Trigonometric Functions Table Summary.

27:05 d/dx (Sin(x) )= Cos (x) as a graph.

28:30 d/dx (Cos(x)) = Sin (x) as a graph.

29:30 Example: Find d^2/ dx^2 of (Sec (x)).

The Derivative as a Function.

0:00 The Derivative as a Function.

2:27 Example, Sketch the graph of the derivative function using the graph of f(x), with six different graphs.

12:36 Theorem: Differentiation Implies Continuous and its Proof.

16:47 Theorem: Alternative Version. f(x) is not continuous implying that f(x) is not differentiable. With four examples showing when a function f(x) is not differentiable. Discontinuity, Corner points, Cusp " Sharp Points", Vertical Tangents.

23:11 Example Find the derivative of f(x) for the given function f(x)= 1/sqrt(x) and find the derivative at x= 25 in detail.

Limits Part IV. Continuity.

0:00 Limits Part IV. Calculus I. Continuity.

2:54 Continuity Checklist.

4:27 Example with a graph.

9:33 Continuity on an Interval.

10:31 Polynomials and Rational Functions.

11:37 Continuity at Endpoints. Left-Continuous and Right-Continuous. 14:38 Trig Functions.

18:19 Example, y = [2 x^2 +3x + 1] / [ x^2 +5x].

20:01 Example, Piecewise Function.

22:54 Example, y = [ x^5 + 6x +17 ] / [ x^2 -9 ].

24:21 Example, y = [ ( 2x + 1)/( x) ]^3 .

25:20 Example, Piecewise Function.

Limits Part I. Covering 10 examples and the Squeeze Theorem.

0:00 Calculus I Limits Part I with graphs and definitions. Watch my second and third videos about limits at: Limits Part II: https://youtu.be/75FzRUC48Ng Limits Part III: https://youtu.be/I7GO24j0W2g

1:54 Average Velocity.

5:27 Instantaneous Velocity.

13:03 Definition of Limits.

14:10 One-sided limits: Limits from the right, Limits from the left.

16:56 Finding limits from graphs.

23:07 Finding limits Algebraically.

26:33 Finding limits by direct substitution.

27:02 Finding limits by simplifying first when you end up dividing by zero. 29:26 Piecewise functions.

30:57 The Squeeze Theorem, known as the Sandwich Theorem.

34:52 Simplify before you plug in h=0.

Average Velocity. A ball is thrown in the air with a velocity 20 ft/sec. its height in feet t seconds later is given by y = 20t - 6 t².

Find the average velocity over the time interval [1, 3].

Graphing Functions using Derivatives.

0:00 Graphing Functions using Derivatives. X-Intercepts.

First Derivative Test. Decreasing. Increasing. Critical Points. Local Min and Local Max. Second Derivative Test. Concave Up. Concave Down. Inflection Points.

0:25 Example: f(x) = (x-3)(x+3)^2.

6:29 Example: f(x) = x^3-6x^2+9x.

9:32 Example: f(x) =x^4-6x^2.

How can you eliminate the discontinuity of f? In other words, how would you set f(5) to ensure f is continuous at 5?

Evaluate the following Integrals.

Curve Sketching. Inc. & Dec. Concave Up. Concave Down, Abs Max, and Abs Min. The Extreme Value Theorem.

A man 6-ft tall walks away from a streetlight mounted on a 15-ft tall pole at a rate of 5-ft/s.

Epsilon and delta Explained with an Example in Calculus.

Limit sin 3x over x

Substitution Rule for Indefinite and Definite Integrals.

0:00 Substitution Rule for Indefinite and Definite Integrals.

0:43 Substitution Rule for Indefinite Integrals.

3:57 Substitution Rule for Definite Integrals.

4:24 Example: Integrate ( x + 1 ) ^ 12.

7:43 Example: Integrate 3 times e ^ ( 3x +1).

11:41 Example: Integrate 2x times ( x^2 - 1 )^ 99.

16:51 Example: Integrate (Sin ^ (x)) times (Cos (x)).

0:00 Definite Integral and General Riemann Sum.

2:23 Example: f(x) = 2x-6. 25:25 Example: y = sin (x) from 0 to 2pi.

7:50 Example: integration of a semicircle function with a diameter of 8, sitting on the x-axis.

10:16 Examples of Properties of Integrals.

10:16 Example 1.

14:30 Example 2.

17:03 Bonus 1.

21:03 Bonus 2.

23:01 Example 3.

L'hopital's Rule

0:00 L'hopital's Rule. Indeterminate Form 0/0, inf/inf, 0 times inf, inf – inf, 0^0.

5:12 Example: lim ( X^3 + x^2 -2x )/( x – 1) as x approaches 1.

6:52 Example: lim (e^x -x – 1) /(x^2) as x approaches 0 “repeated “.

8:54 Example: lim (4 x^3 -6 x^2 +1) /(2 x^3 -10x +3) as x approaches infinity. 10:40 Example: lim ( ln x)/ (csc x) as x approaches 0+.

13:50 Example: lim (x^2) times sin (1/4 x^2) as x approaches infinity.

16:47 Example: lim ( x – sqrt (x^2 -3x)) as x approaches infinity.

21:59 Example: lim x^x as x approaches 0+.

What Derivatives Tell Us.

0:00 What Derivatives Tell Us. Increasing and decreasing. Critical Points. Concave up and concave down. Inflection Points.

10:08 Example in detail.

Maxima and Minima.

0:00 Maxima and Minima.

0:33 Absolute Maxima and Minima.

2:08 Examples within Graphs.

12:21 Extreme Value Theorem.

13:02 Local Maximum and Local Minimum.

13:59 Local Extreme Value Theorem.

14:44 Critical Points.

16:57 Example: Find the critical points of the following function f(x) = 3x^2- 4x +2.

18:00 Example: Find the critical points of the following function f(x) = (1 / x ) +ln (x).

18:59 Example: Find the critical points of the following function f(x) = x ^2 (SQRT ( x+5)).

20:35 Example: Determine the location and value of the Absolute Extreme Values of f ( x ) = x^2 -10 on [ -2, 3].

22:57 Example: Determine the location and value of the Absolute Extreme Values of f ( x ) = x^4 -4 x^3 +4 X^2 on [ -1, 3].

Average Cost and Marginal Cost.

0:00 Average Cost and Marginal Cost.

1:32 Example I.

4:29 Example 2.

Bonus.

The Product and the Quotient Rules.

0:00 The Product and the Quotient Rules.

4:04 Two simple Examples. f(x) = (x)(x^3+1) and f(x) = (x^3 + 2x) / x.

8:36 Example on the Product Rule. f(x)= 2x^6 (3x^2 +1).

10:43 Example on the Quotient Rule. f(x) = (1+x)/(3+2x).

12:11 Example: f(x) = 7x – 2x e^x

13:18 Find the equation of the tangent line of f(x)=(x+4)/(x-1) at x = 3.

16:02 Population Growth: Two Examples with Graphs.

Introduction to Derivative.

0:00 Introduction.

This video covers the main two limit formulas applying the limit as x approaches a to the " slope formula" between two distinct points. Taking everything from the slope of the Secant to the slope of the Tangent at x = a. And that is the derivative of the function "f" at x = a. By applying the limit, we move from the Average velocity between two distinct points to the Instantaneous Velocity at a single point.

12:12 Example 1.

19:08 Example 2.

23:26 Example 3.

28:32 Example 4.

Limits Part III. Limits at Infinity. See the Description below for details.

0:00 Calculus I Limits Part III Watch my other video: Limits Part I at https://youtu.be/aqs2IB28SW4 and Limits Part II at https://youtu.be/75FzRUC48Ng Limits at Infinity. End Behavior. Vertical Asymptote. Horizontal Asymptote. The Squeeze Theorem. Slant Asymptote. [Using Long Division]. Introduction with the limit of x^n when n is Odd and when n is Even as x approaches infinity. End Behavior.

5:10 Example, y = [( 2+( 1/ x^2)]

6:44 Example, y = [ 5 + (sin x / sqrt (x)) ]. Using the squeeze Theorem.

10:37 Example, y = (3 x^ 4 - 6 x^2 +x -10 ). Using the Leading Term Concept. 12:23 Example, y = ( - 2 x^3 +3 x^2 -12 ). Using the Leading Term Concept. 14:13 End Behavior of Rational Functions. Example, y = ( 3 x +2 ) / ( x^2 -1).

17:07 Example, y = [ 40 x^4 + 4 x^2 -1 ] / [ 10 x^4 + 8 x^2 +1 ].

18:36 Example, y = [ x^3 -2x +1] / [ 2 x +4]. 20:44 Summary.

22:16 Slant Asymptote [ Using Long Division]. y = [ 2 x^2 +6 x - 2] / [x + 1]. 25:07 End behavior of exponential functions, Logs, Sine, and Cosine.

28:40 MORE EXAMPLES. Horizontal Asymptote. Vertical Asymptote.

Use the Definition of the Derivative to find f ' (x).

The Derivative Formula I

Limits at infinity.

Use the Intermediate Value Theorem to show that there is a root of the given equation x^2+x-9=0 in the interval (1,2).

Riemann Sum. Right Hand, Left Hand and Midpoint Rule.

Optimization. Find two positive numbers whose Product is 100 and whose Sum is a Minimum.

Implicit Differentiation. Find dy/ dx for 2x^3+x^2y-xy^3=2.

The Quotient Rule Proof in Calculus.

Evaluate the Limit, or state that it does not exist.

Limit as h approaches 0 of [ SQR (9+h) - 3] / h.

Limit with Conjugate

Fundamental Theorem of Calculus Part I and Part II with Examples.

0:00 Fundamental Theorem of Calculus. Area Functions.

3:06 Fundamental Theorem of Calculus Part I.

5:26 Fundamental Theorem of Calculus Part II.

7:37 The Inverse Relationship Between Differentiation and Integration. 10:09 Example 1: Integral of 4 x^3.

11:34 Example 2: Integral of 3 x^2 +2x.

23:35 Example 3: Integral of 8 x^(1/3)

15:33 Example 4: Integral of 2 Cos( x).

17:59 Example 5: Integral of 2/(SQRT (x)).

19:27 Example 6: Integral of (2+ (SQRT (x))/(SQRT (x)).

20:47 Example 7: Integral of e^x.

22:06 Example 8: Integral of (1-x) (x-4).

23:39 Example 9: Integral of 1/((SQRT (1-x^2)).

25:12 Example 10: Integral of (Sec (x) )^2.

25:55 Example 11: Integral of 3/x.

28:51 Example 12: d/dx of the integral of (t^2 + t +1).

30:10 Example 13: d/dx of the integral of (Sin (x) )^2.

30:59 Example 14: d/dx of the integral of Cos (t^2).

36:22 Example 15: d/dx of the integral of (t^4 +6).

37:19 Example 16: d/dx of the integral of ln ( t^2).

Linear Approximation and Differentials.

0:00 Linear Approximation and Differentials. Differentials. Delta y vs dy and delta x vs dx.

17:17 Example: Approximate the Change in f(x) = x^9 -2x + 1 when x changes from 1.00 to 1.05

20:46 Example: Find the Linear Approximation to f(x) = x^3 -5x +3 at x = 2

Optimization Problems.

0:00 Optimization Problems.

First Derivative Test.

Second Derivative Test.

Largest Possible Product.

Write the Objective Function as a Function of One Variable.

Maximum Area Rectangles.

Minimum Sum.

A Rectangular pen is built with one side against a barn.

A rancher plans to make four identical and adjacent rectangular pens against a barn.

Rectangles beneath a Semicircle.

2:37 Example: What two non-negative real numbers with a sum of 30 have the largest possible product.

9:10 Example: Maximum Area Rectangles.

13:02 Example: Minimum Sum.

16:35 Example: A Rectangular Pen is built with one side against a Barn. What dimensions maximize the area of the Pen?

20:21 A Rancher plans to make four identical and adjacent Pens against a Barn.

25:47 Example: Rectangles beneath a Semicircle.

Related Rates

0:00 Related Rates.

0:39 Example One: The sides of a square increase in length at a rate of 2m/sec. At what rate does the area of the square change when the sides are 5 m long?

3:59 Example Two: The area of a circle increases at a rate of 2 cm^2/sec. 7:47 Example Three: A Spherical Balloon is inflated and its volume increases at 8 in ^3/ min.

10:49 A Hot-Air Balloon is 150 ft above the ground when a Motorcycle.

17:30 Example Four: An inverted conical water tank.

24:14 Example Five: A 13-ft Ladder is leaning against a vertical wall.

Derivatives as Rates of Change

0:00 Derivatives as Rates of Change.

4:09 Average Velocity and Instantaneous Velocity.

6:27 Velocity, Speed, and Acceleration.

7:39 Example 1: Position, Velocity, and Acceleration.

16:06 Example 2: Position, Velocity, and Acceleration.

23:12 Bonuses.

25:44 Example 3: A dropped stone on Earth.

28:34 Example 4: A stone is thrown vertically upward.

Rules of Differentiation

0:00 Rules of Differentiation.

0:37 Constant Rule.

3:18 Power Rule.

8:50 Constant Multiple Rule.

9:36 Sum Rule.

10:41 The Derivative of e^x.

13:05 Higher Order Derivatives.

14:21 EXAMPLES.

22:31 Finding Slope Locations.

Limits Part V. Precise Definition of Continuity.

Precise Definition of Continuity. Calculus I.

Corrections, I started saying this is Limits Part IV, but it is Part V.

Covering the Precise Definition of limits with one example. To master the use of this definition, you will need many examples. I will keep that in mind in the future. So, if I make another video with this topic, I will call it Part II.