Cubic Function 2019

Jacob Sichamba Online Math

7 Sept 202222:14

EducationalLearning

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TLDRIn this educational YouTube video, the host, Chamber Jacob, guides viewers through solving a series of math problems related to a given function y = x^3 - 5x + k. He starts by finding the value of k, then moves on to graphing the function within a specific range using a scale. The host also demonstrates how to solve the equation x^3 - 5x = 0 and find the intersecting points with y = 3. He concludes by calculating the area under the curve and finding the equation of the tangent line at a specific point. Throughout the video, he emphasizes the importance of checking work and provides tips for graphing and problem-solving, encouraging viewers to subscribe and engage with the content.

Takeaways

📚 The video is an educational tutorial aimed at helping viewers solve a math exam question.
🔍 The presenter, Chamber Jacob, begins by asking viewers to subscribe to his YouTube channel.
🔢 The first task involves finding the value of 'k' by substituting given values into a mathematical function, resulting in k = -9.
📈 The second task requires viewers to graph the function y = x^3 - 5x + c, with specific instructions on scaling the axes.
📏 The graphing section includes detailed steps on how to plot points and join them to form the graph of the function.
🧩 The third part of the tutorial involves solving the equation x^3 - 5x = 0 by comparing it to the graphed function and finding intersecting points.
📐 The fourth task is to estimate the area bounded by the curve y = 3 and the line x = -2, using a method of counting boxes and rectangles.
📉 The final task is to find the equation of the tangent to the curve at a specific point, involving calculus to determine the slope and the equation of the line.
👨‍🏫 Chamber Jacob emphasizes the importance of checking work and confirming calculations throughout the tutorial.
📞 The video concludes with an invitation for viewers to contact Chamber Jacob for online tuitions, especially for GCE or internal exams.

Q & A

What is the main purpose of the video?
-The main purpose of the video is to guide viewers through solving a series of math problems related to a given function and graphing, as well as finding the value of 'k', graphing the function, solving an equation, estimating an area, and finding the equation of a tangent to the curve.
What is the value of 'k' calculated in the video?
-The value of 'k' is calculated to be -9, after substituting x with -3 in the given function and performing the necessary calculations.
How is the scale used for graphing in the video?
-The scale is used such that 2 centimeters on the x-axis represents one unit and ranges from -3 to 3, while on the y-axis, 2 centimeters represents 5 units and ranges from -10 to 20.
What is the function being graphed in the video?
-The function being graphed is y = x^3 - 5x + c, where 'c' is a constant that was determined to be -9 in the video.
How does the video approach solving the equation x^3 - 5x = 0?
-The video suggests adding 'c' (which is 3 in this case) to both sides of the equation to make it resemble the graphed function and then graphing y = 3 to find the x-intercepts.
What are the x-intercepts found in the video?
-The x-intercepts found are approximately x = -2.2, x = 0, and x = 2.2, which are the points where the graph intersects the line y = 3.
How is the area under the curve estimated in the video?
-The area is estimated by dividing the region under the curve into rectangles and summing the areas of these rectangles, considering the parts that have been cut off.
What is the final answer for the equation of the tangent to the curve at x = -1?
-The final answer for the equation of the tangent is y = 6x + 7, after finding the gradient (derivative) and the point of tangency.
What does the video suggest for those who need online tuitions?
-The video suggests that viewers who need online tuitions can contact the presenter through a provided number for more information on joining online classes.
How does the video encourage viewer engagement?
-The video encourages viewer engagement by asking viewers to subscribe to the channel, hit the notification bell, like, comment, and reach out for online tuitions.

Outlines

00:00

📚 Introduction and Request for Subscription

The script begins with the host, C Chamber Jacob, welcoming viewers to his YouTube channel and encouraging them to subscribe if they haven't already. He then introduces the purpose of the video, which is to revise an exam question together. The first task involves finding the value of 'k' using a given function and substituting the value of 'x' to solve for 'k'. After calculating, the host finds that the value of 'k' is -9, emphasizing the importance of confirming work for accuracy.

05:01

📈 Graphing a Function with Given Scale

The host proceeds to the next task, which is graphing the function y = x^3 - 5x + c, with a specific range provided for both the x and y axes. He explains the scale, which translates to 2 centimeters representing one unit on the x-axis and 2 centimeters representing five units on the y-axis. He demonstrates how to plot points based on given coordinates and then how to join these points to form the graph. The process includes understanding the scale, plotting individual points, and then connecting them to complete the graph.

10:01

🔍 Analyzing the Graph to Solve an Equation

The script continues with solving the equation x^3 - 5x = 0 by comparing it to the previously graphed function. The host points out the need to adjust the equation to match the graph's function by adding '+3' to both sides, resulting in y = x^3 - 5x + 3. He then identifies the x-values where the graph intersects with the line y = 3, which are approximately -2.2, 0, and 2.2. The host advises that slight variations in the x-values are acceptable as long as they are close to the original answer.

15:02

📐 Estimating the Area Bounded by a Curve and a Line

The fourth task involves estimating the area bounded by the curve y = x^3 - 5x + 3 and the line y = 3, with x ranging from -2 to 2. The host describes how to draw the lines that define the region of interest and then calculates the area by considering the rectangles formed within this region. He identifies four rectangles that have been 'cut' and calculates the area by considering the full and partial rectangles, resulting in an area of 10 square units.

20:02

📝 Finding the Equation of a Tangent to a Curve

The final task is to find the equation of the tangent to the curve at a specific point. The host explains the process of finding the derivative of the function, which represents the slope of the tangent line. By substituting x = -1 into the derivative, he finds the slope (gradient) to be 6. Using the point-slope form of a line equation, and knowing the x and y coordinates of the point of tangency, he solves for 'c' and finds the equation of the tangent line to be y = 6x + 7. The host concludes by inviting viewers to like, comment, and subscribe for notifications, and offers online tuition services for those preparing for exams.

Mindmap

Keywords

💡YouTube Channel

A YouTube Channel is a platform where content creators share videos on various topics. In the script, the host, Chamber Jacob, encourages viewers to subscribe to his YouTube channel to receive updates on new content. This is a common practice among content creators to grow their audience and increase engagement.

💡Exam Question

An exam question is a problem or query posed to test a student's knowledge or understanding of a particular subject. In the video, Chamber Jacob is helping viewers to revise and understand an exam question related to finding the value of 'k' in a mathematical function, which is central to the educational theme of the video.

💡Value of k

In mathematics, 'k' often represents a variable or an unknown value that needs to be determined. The script describes a process for calculating the value of 'k' by substituting given values into a function. This is a fundamental concept in algebra and is essential for solving equations.

💡Function

A function in mathematics is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The script mentions a specific function, 'y = x^3 - 5x + k', which is used to demonstrate how to find the value of 'k' and to graph the function.

💡Graph

In the context of mathematics and the script, a graph is a visual representation of the relationship between two or more variables. Chamber Jacob explains how to graph the function 'y = x^3 - 5x + k' using a specific scale and range, which is a key step in understanding the behavior of the function.

💡Scale

Scale in graphing refers to the ratio or proportion used to represent the values of variables on the axes. The script provides a detailed explanation of how to use a given scale to represent units on the x and y axes, which is crucial for accurately plotting points and drawing the graph of the function.

💡Coordinates

Coordinates are pairs of numbers that determine a point's location on a graph. In the script, Chamber Jacob identifies specific coordinates derived from the function to plot on the graph, such as 'negative 3 comma negative 9', which means the point is located at x = -3 and y = -9 on the Cartesian plane.

💡Derivative

The derivative in calculus is a measure of how a function changes as its input changes. In the video, Chamber Jacob calculates the derivative of the function to find the slope of the tangent line at a specific point, which is an essential step in determining the equation of the tangent.

💡Tangent

A tangent to a curve at a given point is a straight line that touches the curve at that point without crossing it. The script describes the process of finding the equation of the tangent to the curve of the function at x = -1, which involves using the derivative to find the slope and then using a point on the curve to find the equation of the line.

💡Area

In the script, Chamber Jacob discusses how to estimate the area bounded by the curve 'y = x^3 - 5x + k' and the horizontal line 'y = 3'. This involves calculating the area of rectangles formed between the curve and the line within a specified range on the x-axis, which is a common problem in calculus and geometry.

Highlights

Introduction to the YouTube channel by Chamber Jacob.

Request for subscription to the YouTube channel.

Explanation of how to find the value of 'k' in a given function.

Mistake in calculating the value of 'k' and correction to it.

Importance of confirming calculations to avoid errors.

Instructions on graphing a function with a given scale.

Process of plotting points and joining them to form a graph.

Solving an equation by comparing it with the graph of a function.

Identifying intersecting points on a graph.

Estimating the area bounded by a curve and a horizontal line.

Method to calculate the area of rectangles to find the total area under a curve.

Finding the equation of a tangent to a curve at a specific point.

Deriving the derivative of a function to find the gradient of a tangent.

Using the point-slope form to find the equation of a line.

Invitation to like, comment, and subscribe for new content notifications.

Offer for online tuitions and contact information for inquiries.

Transcripts

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