[ASMR] THIS is What a University MATH Exam Looks Like

The speaker welcomes viewers to an ASMR video focused on a university math exam. They explain that the video will showcase what a university exam looks like and involve solving math problems from the exam. The speaker encourages viewers to like the video and subscribe to the channel, mentioning their goal of reaching 100,000 subscribers. They discuss their inspiration for the video, which stems from their experience studying maths at university and wanting to share the乐趣 (fun) of math with the audience. The speaker provides an overview of what university entails, particularly in the UK, and introduces the specific papers they will cover: a calculus paper and a general mathematics introductory course.

The speaker delves into the specifics of the calculus paper, explaining the structure of exams in Scottish universities. They detail the balance between exam and coursework marks, the typical time allotted for exams, and the format of the calculus paper, which includes eight questions worth 10 marks each. The speaker also discusses the second paper, which covers a range of topics including combinatorics, algebra, matrices, and geometry. They express hope that high school students finding the video relevant and helpful for their own math studies.

The speaker begins solving the first question from the calculus paper, which involves functions f and g. They explain the process of finding f minus g and f composed with g, using the sine function and x cubed minus five. The speaker then moves on to questions about the domain and range of f, and limits involving x squared plus expressions. They demonstrate the steps and reasoning involved in solving these calculus problems, aiming to make the process clear and understandable for the viewers.

Continuing with the calculus paper, the speaker tackles more questions on limits, including evaluating expressions as x tends to one and working with sine squared x plus cos squared x. They apply mathematical concepts such as factorization and the use of L'Hôpital's rule to solve the limit problems. The speaker also touches on the topic of function continuity, explaining the conditions for a function to be continuous at a point and applying this to the given function.

The speaker transitions to questions involving differentiation and integrals. They discuss the chain rule for finding the derivative of a composite function, such as cos of sine x, and the product rule for differentiating a function like x squared e to the minus two x. The speaker also addresses a differential equation, showing how to verify if a given function y equals x squared e to the minus two x is a solution. They demonstrate the process of differentiating the function and simplifying the result to show that it equals zero, confirming the solution.

The speaker tackles integration problems, starting with finding the area between the sine curve and the y-axis for x between zero and pi. They correctly calculate the area using the integral of sine x and evaluate it to get two square units. Next, they approach an integration by parts problem, evaluating the integral of x e to the x with the help of the integration by parts formula. The speaker simplifies the expression and arrives at the correct result, demonstrating their proficiency in calculus techniques.

The speaker moves on to a more general mathematics paper, covering topics like combinatorics, complex numbers, and matrices. They calculate the number of outcomes with exactly five heads when a coin is tossed 15 times using combinatorics, and deal with complex numbers by rationalizing the denominator and expressing the result in the form a plus bi. The speaker also solves a system of linear equations using elementary row operations, leading to the values of a, b, and c.

In the final part of the video, the speaker calculates the vector product of two given vectors a and b. They explain the process of finding the cross product and computing the determinant to arrive at the answer. The speaker wraps up the video by thanking the viewers for their support and expressing their enjoyment in creating math-related content. They encourage viewers to provide feedback and suggest topics for future videos.