Category Archives: MA 1505 Mathematics I

Prediction of Final Exam 2014-2015 Semester I

Module:                 MA 1505 Mathematics I

Time:                      2 hours ( 120 minutes ), Saturday, 22-Nov-2014 (Morning)

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 convergence domain 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:

http://www.wolframalpha.com

z=x^{2}+y^{2}, z=-(x^{2}+y^{2})              infinite paraboloid

z=x^{2}-y^{2}             hyperbolic paraboloid

(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=R^{2}  sphere with radius R>0 and center (x_{0},y_{0},z_{0})

x^{2}+y^{2}=R^{2},  y^{2}+z^{2}=R^{2}, z^{2}+x^{2}=R^{2}       cylinder

ax+by+cz=d, \text{ where } a,b,c,d \in \mathbb{R}             Plane

y=x^{2}+c \text{ and } x=y^{2}+c, \text{ where } c\in \mathbb{R}             Parabola

Chapter 6: Fourier Series:

Fourier Coefficients of functions with period 2\pi: 1 question. Especially, a_{2014} and b_{2014}  (Integration by parts).

Fourier Coefficients of functions with period 2L: 1 question, where L is a positive real number. Especially, a_{2014} and b_{2014} (Integration by parts).

Calculate the summation of Fourier coefficients: 0-1 question. Especially, \sum_{n=0}^{\infty} a_{n} and \sum_{n=1}^{\infty} a_{n}.

Cosine and sine expansion of function on the half domain: 1 question.

Chapter 7: Function of Several Real Variables

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 Integral

Double integral, polar coordinate: 1 question. (The formula of polar coordinate in the plane).

Reverse the order of integration of double integral: 1 question. (Draw the picture of domain R and reverse the order of dx and dy).

Volume of the solid: 1 question. (Double integral, find the function z=z(x,y) and the domain R on the xy-plane. If the domain R is a disk or a sector, then you can use the polar coordinate).

Area of the surface: 1 question. (Partial Derivatives of functions with two variables, the domain R on the xy-plane. If the domain R is a disk or a sector, then you can use the polar coordinate. The area of a surface is a special case of the surface integral of a scalar field).

Triple integral: 0-1 question. (The method to calculate the triple integral is similar to double integral).

Chapter 9: Line Integrals

Length of a curve: 0-1 question. (Parameter equation of the curves. Length of a curve is a special case of line integral of a scalar field).

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. When the value of a line integral of vector field is independent to the curve C, where C has the fixed initial point and the terminal point?).

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 and left hand rule.

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 z=z(x,y) and the projection of the surface on the xy-plane, Cross product of vectors, the length of vectors. Change the surface integrals of scalar fields to double integrals).

Surface integrals of vector fields: 1 question. (The equation of surface z=z(x,y) and the projection of the surface on the xy-plane, Cross product and Dot product of vectors).

Stokes’ Theorem: 1 question. (This is a rule on line integrals of vector fields and surface integrals of vector fields. Remember the operator curl. Pay attention to the orientation of the curve on the boundary, i.e. the right hand rule).

Divergence Theorem: 0-1 question. (This is a rule on surface integrals of vector fields and triple integrals. Remember the operator div).

Prediction of Middle Term Test

Module:                 MA 1505 Mathematics I

Time:                     1 hours ( 60 minutes )

Questions:            10 Multiple Choice Questions.

Average speed:     6 minutes per question.

Scores:                  20% in final score.

The contents in high school:

Trigonometric functions, some basic inequalities and identities.

Questions in middle term test:

Question 1. Derivatives, Tangent line of a function, Intersection point of tangent line and x-axis, y-axis. Basic Rules of differentiation, Chain Rule.

Question 2. Critical points of a function, how to calculate the maximum and minimum value of a function.

Question 3. Integration by parts, integrate trigonometric functions.

Question 4. Fundamental theorem of calculus.

Question 5. Find the area which is bounded by some curves.

Question 6. Mathematical models. ( e.g. light and ball drop, ship and so on).

Question 7. Radius of convergence of a power series, the interval of convergence of a power series.

Question 8. Calculate the Taylor series of functions, Calculate the coefficients of Taylor series.

Question 9. How  to use Taylor series to calculate the solution of an equation.

Question 10. How to use Taylor series to calculate the summation of some series. ( Integration and differentiation).

Question 11. The length of a curve, the tangent line of a curve.

Question 12. Dot product and cross product of two vectors, equation of planes, normal vector of a plane, distance between a point and a plane.

MA 1505 Tutorial 1: Derivative

Definition of Derivative:

f^{'}(x)=\lim_{\Delta x\rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}

Rule: Assume f(x) and g(x) are two differentiable functions, the basic rules of derivative are

(f\pm g)^{'}(x)=f^{'}(x)\pm g^{'}(x)

(f\cdot g)^{'}(x)= f^{'}(x) g(x) + f(x)g^{'}(x)

(f/g)^{'}(x)=(f^{'}(x)g(x)-f(x)g^{'}(x))/(g(x))^{2}

(f\circ g)^{'}(x)=f^{'}(g(x))g^{'}(x)

Definition of Critical Point: x_{0} is called a critical point of f(x), if f^{'}(x_{0})=0.

If f^{'}(x)>0 on some interval I, then f(x) is increasing on the interval I. Similarly, if f^{'}(x)<0 on some interval I, then f(x) is decreasing on the interval I.

Tangent Line: Assume f(x) is a differentiable function on the interval I, then the tangent line of f(x) at the point x_{0}\in I is y-f(x_{0})=f^{'}(x_{0})(x-x_{0}), where f^{'}(x_{0}) is the slope of the tangent line.

Derivative of Parameter Functions: Assume y=y(t) and x=x(t), the derivative y^{'}(x) is y^{'}(t)/x^{'}(t), because the Chain Rule of derivatives.

Question 1. Calculate the tangent line of the curve x^{\frac{1}{4}} + y^{\frac{1}{4}}=4 at the point (16,16).

Method (i). Take the derivative of the equation x^{\frac{1}{4}}+y^{\frac{1}{4}}=4 at the both sides, we get

\frac{1}{4}x^{-\frac{3}{4}} + \frac{1}{4}y^{-\frac{3}{4}} y^{'}=0.

Assume x=y=16, we have the derivative y^{'}(16)=-1. That means the tangent line of the curve at the point (16,16) is y-16=-(x-16). i.e. y=-x+32.

Method (ii). From the equation, we know y(x)=(4-x^{\frac{1}{4}})^{4} , then calculating the derivative directly. i.e.

y^{'}(x)=4(4-x^{\frac{1}{4}})^{3}\cdot (-1)\cdot \frac{1}{4}x^{-\frac{3}{4}}

Therefore, y^{'}(16)=-1.

Method (iii). Making the substitution x=4^{4}\cos^{8}\theta, y=4^{4}\sin^{8}\theta, then (16,16) corresponds to \theta=\pi/4. From the derivative of the parameter functions, we know

\frac{dy}{dx}= \frac{dy/d\theta}{dx/d\theta}=\frac{4^{4}\cdot 8\sin^{7}\theta\cdot \cos\theta}{4^{4}\cdot 8\cos^{7}\theta\cdot (-\sin\theta)}

If we assume \theta=\pi/4, then y^{'}(16)=-1.

Method (iv). Geometric Intuition. Since the equation x^{\frac{1}{4}}+y^{\frac{1}{4}}=4 is a symmetric graph with the line y=x, and (16,16) is also on the symmetric line. Therefore, the slope of the curve at the point (16,16) is -1. Hence, the tangent line is y=-x+32.

Question 2. Let y=(1+x^{2})^{-2} and x=\cot \theta. Find dy/dx and express your answer in terms of \theta.

Method (i). y=\frac{1}{1+x^{2}}= \sin^{2}\theta

\frac{dy}{dx}=\frac{dy/d\theta}{dx/d\theta} = \frac{2\sin\theta \cos \theta}{-\sin^{-2}\theta}= - \sin^{2}\theta\sin2\theta.

Method (ii). \frac{dy}{dx}=-\frac{2x}{(1+x^{2})^{2}} = -\frac{2\cot \theta}{(1+\cot^{2}\theta)^{2}}=-\sin^{2}\theta\sin 2\theta.

Prediction of Final Exam 2013-2014 Semester I

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:

http://www.wolframalpha.com

z=x^{2}+y^{2}             infinite paraboloid

z=x^{2}-y^{2}             hyperbolic paraboloid

(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=R^{2}  sphere with radius R and center (x_{0},y_{0},z_{0})

x^{2}+y^{2}=R^{2}          cylinder

ax+by+cz=d, \text{ where } a,b,c,d \in \mathbb{R}             Plane

y=x^{2}+c \text{ and } x=y^{2}+c, \text{ where } c\in \mathbb{R}             Parabola

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).

MA 1505 Tutorial 11: Surface Integral, Divergence Theorem and Stokes’ Theorem

Surface Integrals of Scalar Fields: Assume f: U \subseteq \mathbb{R}^{3} \rightarrow \mathbb{R} is a function, r: D\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} is a surface S. Then the surface integral is

\iint_{S} f dS= \iint_{D} f(\textbf{r}(x,y)) || \textbf{r}_{x} \times \textbf{r}_{y} || dxdy

where the left hand side is the surface integral of the scalar field and the right hand side is the multiple integration. \textbf{r}_{x} \times \textbf{r}_{y} denotes the cross product between \textbf{r}_{x} and \textbf{r}_{y} ,

|| \textbf{r}_{x} \times \textbf{r}_{y} ||  denotes the length of the vector \textbf{r}_{x} \times \textbf{r}_{y}.

Remark.  If f(x,y,z)=1 for all (x,y,z) \in \mathbb{R}^{3} , and r: D\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} is a surface, then

the left hand side is \iint_{S} dS = \text{ the surface area of } S.

the right hand side is \iint_{D} \sqrt{1+(f_{x})^{2}+ (f_{y})^{2} } dxdy , since \textbf{r}(x,y)=(x,y,f(x,y)), \text{ where } (x,y) \in D, \textbf{r}_{x}=(1,0,f_{x}) and \textbf{r}_{y}=(0,1,f_{y}), the cross product \textbf{r}_{x} \times \textbf{r}_{y}= (-f_{x}, -f_{y},1).

That means:

\text{ the surface area of } S= \iint_{D} \sqrt{1+(f_{x})^{2}+(f_{y})^{2} }dxdy.

Surface Integrals of Vector Fields:

Imagine that we have a fluid flowing through S, such that \bold{F}(x) determines the velocity of the fluid at \bold{x}. The flux is defined as the quantity of fluid flowing through S per unit time.

This illustration implies that if the vector field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S, and neither in nor out. This also implies that if \bold{F} does not just flow along S, that is, if F has both a tangential and a normal component, then only the normal component contributes to the flux. Based on this reasoning, to find the flux, we need to take the dot product of \bold{F} with the unit normal vector to S at each point, which will give us a scalar field, and integrate the obtained field as above.

1280px-Surface_vectors

Assume \textbf{F} : U \subseteq \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} is a vector field, r: D\subseteq \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} is a surface S. Then the surface integrals of the vector field F is

\iint_{S} \textbf{F} \cdot d \textbf{S} = \iint_{S} \textbf{F} \cdot \textbf{n} dS

The left hand side is the surface integral of vector field and the right hand side is the surface integral of scalar function, since \textbf{F} \cdot \textbf{n} is a scalar function. That means,

\iint_{S} \textbf{F} \cdot d \textbf{S} = \iint_{S} \textbf{F} \cdot \textbf{n} dS = \iint_{D} \textbf{F}( \textbf{r}(x,y)) \cdot ( \textbf{r}_{x} \times \textbf{r}_{y}) dxdy

Divergence Theorem (Gauss’s theorem or Ostrogradsky’s theorem)

This theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.

\iint_{S} \textbf{F} \cdot d \textbf{S} = \iiint_{V} \nabla \cdot \textbf{F} dV = \iiint_{V} (\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}) dxdydz

where V \subseteq \mathbb{R}^{3} is a bounded domain and \partial V=S, \textbf{F}=(P,Q,R) is a vector field.

800px-Divergence_theorem.svg

Stokes’ Theorem

\int_{\partial \Sigma} \textbf{F} \cdot d\textbf{r} = \iint_{\Sigma} ( \textbf{curl F} ) \cdot d \textbf{S}

where \textbf{curl F}= (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}) \textbf{i} + (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}) \textbf{j} + ( \frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}) \textbf{k} is a vector field. \Sigma is a compact surface and \partial \Sigma  is the boundary of \Sigma. The curve \partial\Sigma has the positive orientation, that means following the right hand rule.

429px-Stokes'_Theorem.svg