A-level Maths

Lecturer explores fundamental trigonometric identities, starting with Pythagorean trigonometry (sin²θ + cos²θ = 1) from GCSE, derived using a unit circle and right triangle. They then define reciprocal trig functions and expand upon the initial identity to derive two more: cot²θ = cos²θ and tan²θ = sin²θ/cos²θ. Subscribe to @AxiomTutoringCourses for more advanced trigonometry lessons.

In this A-level trigonometry video, we delve into the essential compound angle formulae. You'll learn how to state the formulae for sine and cosine of sums and differences of angles. We'll explore how the odd and even properties of sine and cosine functions allow us to derive new expressions. This video will equip you to understand and derive formulae for sine, cosine, and tangent of A plus B and A minus B. Subscribe to @AxiomTutoringCourses for more A-level math tutorials.

Welcome to the third video in our A-level trigonometry series, where we build upon compound angle formulae. In this session, we will derive and understand the double angle formulae for sine, cosine, and tangent. We will explore how to express sine of 2x as 2 sine x cos x, and uncover the three distinct forms of the cosine of 2x identity. Subscribe to @AxiomTutoringCourses

In this A-level trigonometry video, we delve into the crucial skill of proving trigonometric identities. You'll learn how to strategically use known identities to manipulate and simplify expressions, transforming one side of an identity to match the other. We'll cover the proper use of equivalence notation and demonstrate working through examples, starting with a fundamental identity involving secant squared and then tackling a more complex triple angle identity. Subscribe to @AxiomTutoringCourses for more A-level math tutorials.

In this video, we delve into the crucial skill of proving trigonometric identities. Learn how to effectively use known trigonometric identities to manipulate and simplify expressions, transforming one side of an identity to match the other. We'll introduce the equivalence notation and emphasize the distinction between proving identities and solving equations. Through clear examples, you'll master the process of starting with the left-hand side and systematically working towards the right-hand side. Subscribe to @AxiomTutoringCourses for more A-level trigonometry tutorials.

In this A-level trigonometry video, we explore the graphs of sine, cosine, and tangent functions. We'll examine their key features, including maximum and minimum points, periods, and symmetry. Understanding these graphical characteristics is crucial for mastering trigonometric concepts. This video covers the graphs of y = sin x, y = cos x, and y = tan x within the domain of -2π to 2π. We'll identify their unique properties, such as rotational symmetry for sine and tangent, and y-axis symmetry for cosine. We'll also discuss the concept of periodicity and the presence of asymptotes in the tangent graph. Don't forget to subscribe to @AxiomTutoringCourses for more math tutorials.

This video tutorial teaches you how to sketch the graphs of trigonometric functions: sine x, cosine x, and tangent x. We'll cover their key features including x-intercepts, maximum and minimum values, and vertical asymptotes for tangent x. The graphs will be plotted over the domain of minus 2 pi to 2 pi to showcase their full shape and periodicity. Learn how to identify and generalize these features for each function. Subscribe to @AxiomTutoringCourses for more math tutorials.

This video explains how to find exact trigonometric values using the unit circle and special triangles. We'll cover recalling exact values for key angles in radians: 0, pi/6, pi/4, pi/3, and pi/2. Understanding these values is crucial for solving trigonometric equations, proving identities, and avoiding calculator rounding errors. The tutorial walks through deriving these values from a right-angled isosceles triangle and a 30-60-90 triangle. Subscribe to @AxiomTutoringCourses for more math tutorials.

In this video, we begin our A-level trigonometry series by tackling trigonometric equations. We'll learn to identify the structure of these equations and apply appropriate methods, focusing on using exact trigonometric values. The video also demonstrates how to utilize trigonometric graph properties to find all solutions within a specified interval, using examples within 0 to 2 pi. We'll work through solving sine x equals a half, 2 sine theta equals 3 cos theta, and 2 sine squared theta equals cos theta plus 1, illustrating techniques like using inverse functions, recognizing tangent identities, and solving quadratic equations derived from trigonometric substitutions. Subscribe to @AxiomTutoringCourses for more A-level math tutorials.

This video continues our A-level trigonometry theory by focusing on solving trigonometric equations. We'll explore how to find solutions within specific intervals and how to derive general solutions when no interval is given. The tutorial covers solving equations with compound angles, including a detailed example of transforming a quadratic into a solvable form and finding the roots. We also demonstrate how to handle equations involving tangent and sine, simplifying them to find all possible solutions. Subscribe to @AxiomTutoringCourses for more A-level math tutorials.

In this video, we embark on part one of differentiating trigonometric functions from first principles in our A Level Mathematics calculus series. We will utilize the definition of the derivative and apply it to simple trigonometric functions, specifically focusing on finding the derivative of sine x. This process involves understanding and using small angle approximations for sine and cosine, which are crucial for deriving the derivative of sine x. We'll explore how the compound angle formula for sine helps us rewrite f(x+h) and then simplify the limit expression using approximations. Subscribe to @AxiomTutoringCourses for more A Level Mathematics tutorials.

In this video, Kristina introduces the fundamental concept of differentiation from first principles, a crucial topic in A-level mathematics calculus. She explains the definition of the derivative as a limit and visually demonstrates its meaning using a curve and two points. The video clarifies why the variable 'h' in the derivative's formula must tend towards zero without ever equalling zero. This foundational understanding is essential for calculating the gradient of a curve at any given point. Subscribe to @AxiomTutoringCourses for more A-level mathematics tutorials.

This video continues our A Level Mathematics calculus series, focusing on differentiation from first principles. We'll work through an example to find the derivative of the function f(x) = 3x^2 using the definition of a derivative as the limit of f(x+h) - f(x) all over h as h approaches 0. We'll expand f(x+h), substitute it into the derivative formula, simplify the expression by canceling terms, and factor out h to find the final derivative. By the end of this lesson, you'll be able to confidently calculate derivatives from first principles for similar functions. Subscribe to @AxiomTutoringCourses for more math tutorials.

This video tutorial continues the A-level mathematics calculator series, focusing on differentiation from first principles. We will work through the second example from the previous video, calculating the derivative of two x cubed using the limit definition. The process involves substituting x plus h into the function, expanding the resulting expression, and simplifying. Finally, we will take the limit as h approaches zero to find the derivative. Subscribe to @AxiomTutoringCourses for more math tutorials.

Welcome to the first video in the Calculus A-level Mathematics series, where we begin by defining the gradient of a curve. We'll start by recalling the gradient of a straight line, a fundamental concept from GCSE. Then, we'll delve into defining the gradient of a curve at a specific point using the concept of tangents. A tangent is a straight line that touches a curve at exactly one point. To find the gradient of a curve at a given point, we determine the gradient of the tangent line at that precise location. This video will guide you through the process of understanding and calculating these gradients, laying the groundwork for more advanced calculus topics.

Welcome to the second video in our A-level mathematics calculus series, where we'll introduce the fundamental concept of differentiation. This video focuses on defining the derivative of a function f(x) at a specific point x, represented as f'(x). We'll explore what the derivative signifies, understanding it as the gradient function, also expressible as dy/dx. While we won't be calculating derivatives from scratch in this session, we'll lay the groundwork for grasping its meaning and significance as the gradient of the curve at a given x-coordinate, building upon concepts from the previous video and setting the stage for first principles in the next. Subscribe to @AxiomTutoringCourses for more math tutorials.

Hi everyone, I'm Christina and welcome to video 10 in our A Level Mathematics Calculus series. In this video we're differentiating expressions with powers of X and this is part one. So our objectives in this video are just to introduce formally for differentiating powers of X and we have two special cases. The main one we're going to look at here is to recognize that the derivative of the constant function, that's a function where f of X is equal to a constant number A. When we find the derivative of a constant function we always get zero. So let's have a look at the formulae here. So if n is any real number and A is a constant number, for f of X in the form f of X equals X to the power of n. If we want to find the derivative of that function f dash of X, we bring down the n. So we bring down our power n in front, which becomes our coefficient for our derivative. And then we minus one from the index, which we have here. So that's the general formula we follow when we have a function in the form X to the n. Now in our second formula here, we have a function which is A times X to the n. So we have a coefficient of A in front of our X to the n. So when we want to find our derivative function, we keep our constant A in front. And then we just differentiate our X to the n as we did before. So what happens is we bring down our n in front. That's why we get our A times n here. And then we minus one from the index up here. In our third formula here, if f of X is A to the X. So our function here is X to the power of one. So it's just X. We have our coefficient A. If we want to find the derivative f dash of X of this function, it's just A. It's just the constant in front of our X. And our special case here, when our f of X is a constant value A. So that's any constant value. The derivative of a constant function like this is always zero. So in the next video, we're going to apply these to some worked examples.

This video delves into differentiating trigonometric functions from first principles, specifically focusing on the derivative of tan X. We will walk through the process of using the definition of a derivative to find this, recalling the tangent addition formula and applying small angle approximations for tan h. This method allows us to derive the derivative of tan X, which is secant squared X, also known as one over cos squared X. Subscribe to @AxiomTutoringCourses for more A-level mathematics tutorials.

This video, part 2 of our A-level mathematics calculus series, focuses on differentiating trigonometric functions from first principles. We specifically derive the derivative of cos x using the definition of the derivative. The method involves applying the compound angle formula for cos(x+h) and then utilizing small angle approximations for sin h and cos h as h approaches zero. By simplifying the expression, we demonstrate how the derivative of cos x is found to be -sin x directly from first principles. Subscribe to @AxiomTutoringCourses for more A-level calculus tutorials!

This video continues our A-level mathematics calculus series by focusing on differentiating expressions of powers of x. We will apply the foundational formulas introduced in the previous video through practical, worked examples. Specifically, we'll tackle differentiating functions in the form ax^n. Follow along as we break down how to differentiate 6x^4 and 3x^(1/2). Subscribe to @AxiomTutoringCourses for more math tutorials.

This video continues our A-level mathematics calculus series by focusing on differentiating expressions involving powers of x. We work through two detailed examples, starting with a function that requires simplification using index laws before differentiation. The process involves rewriting the expression into a form suitable for the power rule and then applying it term by term. The second example demonstrates differentiating a function that includes a square root, again emphasizing the importance of rewriting it as a power of x. We cover the step-by-step calculation for each derivative, showing how to handle fractions and negative exponents. Subscribe to @AxiomTutoringCourses for more math tutorials.

This video is the ninth in our I level mathematics calculator series, focusing on differentiating trigonometric functions. We will summarize the derivatives of common trig functions and introduce the derivatives of reciprocal trigonometric functions. You'll learn how to find the derivative of sec x, cosec x, and cot x, building on previously established results from first principles. This comprehensive overview ensures you are comfortable with all essential trigonometric differentiation rules. Subscribe to @AxiomTutoringCourses for more math tutorials.

Welcome to part one of our A-level mathematics calculus series, video 14, where we explore differentiating composite functions using the chain rule. This video will teach you how to recognize and apply the chain rule to differentiate functions within functions, such as y equals 2x plus 1 cubed. We'll break down the chain rule definition and walk through a clear example to illustrate its application. Understanding the chain rule is crucial for simplifying complex differentiation problems by treating nested functions step-by-step. Subscribe to @AxiomTutoringCourses for more math tutorials.

In this video, we delve into the concept of second order derivatives in calculus. Kristina explains the definition of a second order derivative and demonstrates how to calculate it. We'll learn that differentiating a function twice, or differentiating its first derivative once, yields the second order derivative. This crucial concept helps us understand the rate of change of the gradient function, a topic we will revisit when discussing turning points. Watch as we work through an example, finding both the first and second derivatives of a given function step-by-step. Subscribe to @AxiomTutoringCourses for more math tutorials.

In this A-level Mathematics video, Christina introduces the concept of composite functions, starting with part one of the series. You will learn the formal definition of a composite function and understand the notation used to represent them, specifically fg of x. The video explains how the output of one function becomes the input for another, breaking down the process with clear examples. Discover how to combine functions like f(x) = 3x and g(x) = 5 + x to find the composite function fg of x. Subscribe to @AxiomTutoringCourses for more A-level Maths tutorials.

In this A-level mathematics video, we delve into composite functions, specifically addressing whether the order of operations matters. We investigate if fg(x) is always equivalent to gf(x). To explore this, we work through a practical example using f(x) = x squared + 1 and g(x) = x + 5. We meticulously calculate both fg(x) and gf(x) to observe their distinct results. This detailed breakdown demonstrates that composite functions are not commutative, meaning the order in which they are applied is crucial to the final outcome. Subscribe to @AxiomTutoringCourses for more A-level mathematics tutorials.

In this A Level Mathematics video, we tackle composite functions with an exam-style question. We begin by evaluating fg of 1 for the functions f(x) = 3x + 4 and g(x) = 2 over x + 3, demonstrating the step-by-step substitution process. Following this, we solve the equation gf of x equal to 6, showing how to form the composite function and then solve the resulting algebraic equation for x. Subscribe to @AxiomTutoringCourses for more A Level Mathematics tutorials.

This A-level mathematics video introduces inverse functions, denoted by f to the minus 1 of x. We explore how an inverse function 'undoes' the original function by swapping inputs and outputs. The crucial concept is that an inverse function only exists if the original function is one-to-one. This means each output must map from exactly one input; otherwise, the inverse would be ambiguous. Understanding inverse functions is essential for solving a variety of mathematical problems. This lesson lays the groundwork by defining the inverse and explaining the necessary condition for its existence, illustrated with clear examples. We will learn to identify functions that have an inverse and how to conceptualize the relationship between a function and its inverse. Subscribe to @AxiomTutoringCourses for more math tutorials.

Welcome to part two of our A-level mathematics series on inverse functions, where we focus on sketching their graphs. In this video, you will learn how to visually represent the inverse of a function by reflecting its graph across the line y=x. We will also formally revisit and solidify your understanding of the domain and range of inverse functions. Discover the relationship between a function and its inverse using the exponential and natural logarithm functions as a prime example. This fundamental concept of reflection across y=x is key to sketching any inverse function. Subscribe to @AxiomTutoringCourses for more A-level math tutorials.

In this A-level mathematics video, we delve into finding the inverse of a function using algebraic methods. Christina walks through the essential steps, starting with setting y equal to f(x). The core of the process involves rearranging the equation to isolate x as the subject. Finally, the resulting expression is rewritten in function notation to represent the inverse. The video also clarifies how the domain of the inverse function relates to the range of the original function. This tutorial provides a clear, step-by-step algebraic approach to determine the inverse of a function, ensuring a solid understanding of the transformation. We will walk through an example to illustrate each stage of the process. Subscribe to @AxiomTutoringCourses for more A-level mathematics tutorials.

This video demonstrates how to differentiate composite functions using the chain rule. We work through a specific calculus example, y equals 5 over x squared plus 4, breaking down each step of the chain rule application. You will learn how to rewrite the function, identify and differentiate the inner and outer functions, and substitute back to find the final derivative in terms of x. This tutorial provides a clear, step-by-step explanation of this essential calculus technique. Visit AxiomTutoring.com for more math resources and subscribe to @AxiomTutoringCourses for ongoing math help.

In this video, we continue our calculus series by focusing on differentiating composite functions using the chain rule, building upon our previous lesson. You'll learn to apply the chain rule effectively through a step-by-step breakdown of a complex example. We will first identify the inner and outer functions, then find the derivatives of each with respect to their respective variables. Finally, we'll combine these derivatives to solve for the derivative of the composite function in terms of x. Visit AxiomTutoring.com for more resources and subscribe to @AxiomTutoringCourses for future calculus lessons.

This video continues our A-level mathematics calculus series, focusing on differentiating composite trigonometric functions. We will apply the chain rule to solve problems involving functions like sine of 3x squared and 3x to the power of 7 minus sine of x squared plus 6. Learn how to break down complex functions into simpler parts and use the chain rule to find the derivative. By the end of this tutorial, you'll be comfortable differentiating composite trigonometric expressions. Visit AxiomTutoring.com and subscribe to @AxiomTutoringCourses for more helpful math tutorials.

In this video, Christina introduces stationary points and turning points in calculus. She defines a stationary point as any point on a curve where the gradient is zero, and explains the method for finding their coordinates by setting the first derivative to zero. The video then provides a clear example using a quadratic function to demonstrate the calculation of a stationary point. Finally, it distinguishes between stationary points and turning points, explaining that while all turning points are stationary, not all stationary points are turning points, depending on the change in gradient. Visit AxiomTutoring.com and subscribe to @AxiomTutoringCourses.

Welcome to part two of our A-level mathematics calculus series on stationary and turning points. In this video, we will learn how to determine the nature of stationary points using the first derivative. We will explore how the slope of a curve changes around a local maximum and a local minimum. Understanding these changes allows us to classify stationary points without needing to sketch the graph. Join us as we break down this essential calculus concept with clear examples and explanations. Visit AxiomTutoring.com and subscribe to @AxiomTutoringCourses for more math tutorials.

Welcome to part 3 of our A-level Mathematics calculus series, focusing on stationary and turning points. In this video, we define points of inflection, which are distinct from turning points. We explore how the gradient's sign remains consistent around a point of inflection, even as the curvature changes. This concept is crucial for understanding the behavior of functions and will prepare you for investigating stationary points using the second derivative. Visit AxiomTutoring.com and subscribe to @AxiomTutoringCourses.

In this A Level Mathematics Calculus video, we delve into determining the nature of stationary points using the second derivative. Building on previous lessons, we'll explore how calculating the second derivative provides a direct method to classify these critical points. This video outlines the step-by-step process, from finding the first derivative and solving for stationary points to differentiating twice to obtain the second derivative. We then demonstrate how to evaluate the second derivative at each stationary point to identify whether it's a local minimum, a local maximum, or requires further analysis using the first derivative test. The next video will feature practical examples to solidify your understanding. Visit AxiomTutoring.com and subscribe to @AxiomTutoringCourses.