Module: MA 1505 Mathematics I
Time: 2 hours ( 120 minutes )
Questions: 8 questions, each question contains two questions. i.e. 16 questions.
Average speed: 7.5 minutes per question.
Scores: 20% mid-term exam, 80% final exam. i.e. Each question in the final exam is 5%.
Remark: Another Possibility: 5 Chapters, each chapter contains 1 big question, and each question contains three small questions, i.e. 15 questions. 8 minutes per question.
The contents in high school:
Trigonometric functions, some basic inequalities and identities.
The contents before mid-term exam: Please review the details of them.
Chapter 2: Differentiation
Derivatives of one variable functions, derivatives of parameter functions, Chain rule of derivatives, the tangent line of the curve, L.Hospital Rule, critical points of one variable, local maximum and local minimum of one variable function.
Chapter 3: Integration
Integration by parts, Newton-Leibniz Formula, the area of the domain in the plane, the volume of the solid which is generated by a curve rotated with an axis.
Chapter 4: Series
Taylor Series and Power Series, radius of convergence of power series, the sum of geometric series and arithmetic series.
Chapter 5: Three Dimensional Spaces
Cross Product and Dot Product of vectors, projection of vectors, the equation of the plane and the line in 3-dimensional space, Distance from a point to a plane, Distance from a point to a line, the distance between two lines in two or three dimensional spaces, the distance between two parallel planes. Intersection points of two different curves.
The contents after mid-term exam: Must prepare them.
By the way, 2-3 questions means at least 2 questions, at most 3 questions. 0-1 question means 0 question or 1 question.
Geometric Graphs in Three Dimensional Space:
sphere with radius R and center
Chapter 6: Fourier Series:
Fourier series, Parseval’s identity: 2-3 questions. ( Integration by parts, calculate the sum of Fourier coefficients, period 2L functions ( where L is a positive real number), calculate the value of some special series from Fourier series, cosine expansion and sine expansion of function on the half domain).
Chapter 7: Multiple Variable Functions
Directional derivatives, partial derivatives, gradient of functions with two or three variables, Chain Rule of partial derivatives: 1-2 questions. (Pay attention to whether the vector is a unit vector or not. If it is not a unit vector, you should change it to a unit vector first, and then calculate the directional derivatives).
Critical points of two variable functions ( saddle point, local maximum, local minimum): 0-1 question. ( Calculate the partial derivatives first, then evaluate the critical points, so we can decide the property of the critical points from some rules).
Lagrange’s method: 0-1 question. ( Calculate the maximum value of functions under some special conditions. Construct the function first, evaluate partial derivatives secondly, and calculate the critical points of the new functions. In addition, if you use inequality “arithmetic mean” is greater than “geometric mean”, then the question will become easier.)
Chapter 8: Multiple Integration
Double integration, polar coordinate: 1 question. ( The formula of polar coordinate in the plane).
Reverse the order of integration of double integration: 1 question. ( Draw the picture of domain R and reverse the order of dx and dy).
Volume of the solid: 1 question. ( Double integrals).
Area of the surface: 1 question. ( Partial Derivatives of two variable functions, Polar Coordinate).
Chapter 9: Line Integrals
Length of the curve: 0-1 question. ( Parameter equation of the curves).
Line integrals of scalar fields: 1 question. ( The equation of line segment, the equation of the circle with radius R, the length of vectors). Geometric meaning: the area of the wall along the curve.
Line integrals of vector fields: 1 question. ( The equation of line segments, the equation of the circle with radius R, Dot product of vectors). Physical meaning: Work done.
Conservative vector fields and Newton-Leibniz formula of gradient vector fields: 0-1 question. ( Definition of conservative vector field and its equivalent condition).
Green’s Theorem: 1 question. ( Two cases: the boundary is open; the boundary is closed. If the curve is open, you should close it by yourself.) Pay attention to the orientation, i.e. anticlockwise.
Chapter 10: Surface Integrals
Tangent plain of a surface: 0-1 question. ( Partial derivatives, Cross product of two vectors, Normal vector of a plane)
Surface integrals of scalar fields: 1 question. ( The equation of surface, Cross product of vectors, the length of vectors).
Surface integrals of vector fields: 1 question. ( The equation of surface, Cross product and Dot product of vectors).
Stokes’ Theorem: 1 question. ( Pay attention to the orientation).
Divergence Theorem: 0-1 question. ( Triple integrals).