# Chapter 2 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 10
Type: Notes

Is there actually a way to “unlock math” for ourselves and learn the subject quickly? It is a universal and undebatable knowledge that mathematics is a darn difficult subject.

In this chapter we will read these topics:

• Name the mathematicians who invented different calculus.
• Define dependent and independent varaibles.
• Define the derivative of a function.
• What is differentiation?
• Differentiation by definition.
• The first principle or AB-Initio Method.
• Write different notations used for derivatives by different mathematicians.
• Find derivative when n is a positive integer.
• Find derivative when n is a negative integer.
• Find derivative when n is zero.
• Find from the definition, the differential coefficient of the given variable.
• Find from the 1st principle.
• Theorems on differentiation
• State and prove the theorem of derivatives.
• State and prove the difference theorem of the derivative.
• State and prove the product theorem of derivatives.
• State and prove the quotient theorem of derivatives.
• State and prove the reciprocal rule of the derivative.
• The Chain Rule.
• A derivative of Parametric equations.
• Define Parameter.
• Define Parametric equations.
• Define explicit and implicit functions.
• A derivative of Implicit Relations.
• Differentiation of trigonometric functions.
• Prove by the ab-initio method.
• Find derivatives from the first principle.
• Derivation of inverse trigonometric functions.
• Differentiation of exponential functions.
• Define the exponential function.
• Define general exponential function.
• Define natural exponential function.
• Differentiation of logarithmic functions.
• Define the general logarithm function.
• Define common logarithm function.
• Define natural logarithm function.
• Differentiation of hyperbolic functions.
• Define hyperbolic functions.
• A derivative of inverse hyperbolic functions.
• Higher derivatives.
• First derivative
• Second derivative
• Third derivative.
• Fourth derivative.
• Nth derivative.
• Higher derivative.
• Series expansions of functions
• Define Maclaurin’s series expansion of a function.
• State and prove Maclaurin’s series expansion of a function.
• Find Maclaurin’s series for sinx.
• State and prove Taylor’s series expansion of a function.
• Geometrical meaning of derivative.
• State the geometrical interpretation of derivative.
• Define the tangent line.
• Prove that geometrically derivative of a function gives the slope of a tangent line.
• Discuss the tangent line to the graph of function |x| at x=0.
• Define increasing function.
• Define a decreasing function.
• Define a constant function.
• Define an increasing function in terms of derivatives.
• Define decreasing function in terms of derivatives.
• Define constant function in terms of derivatives.
• Define stationary point.
• Define critical value and critical point.
• Define turning point of a function.
• Define point of inflection.
• Define relative minima of a function.
• Define relative maxima of a function.
• What is the relative extrema of a function.
• Define first derivative test.
• Applications of maxima and minima.
• Find two positives integers whose sum is 9 and the product of one with the square of the other will be maximum.
• What are dimensions of a box of a square base having largest volume if the sum of one side of the base and height is 12cm.
• The perimeter of a triangle is 20cm. If one side is of the length 8cm, what are length of the other sides for the mum area of the triangle?
• FInd the two positive integers whose sum is 30 and their product will be maximum.

# Chapter 3 (Differentiation) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 8
Type: Notes

In this chapter we will read these topics:

• What is the differential of a function f(x) and what is the differential coefficient?
• What is the difference between delta(y) and dy?
• Use differential to find approximate value.
• Find the approximate increase in the volume of a cube of the length its each edge changes from 5 to 5.02.
• Find the approximate increase in the area of a circular disc if its diameter is increased from 44com to 44.4.
• Define integration.
• Some basic tips for integration.
• Define the indefinite integral.
• Write two theorems on anti-derivatives.
• Some basic formulae.
• Evaluate the problems.
• Write a note on integration by substitution.
• Write some important trigonometric substitutions.
• Integration by parts.
• Integration by partial fractions.
• When denominator contains non-repeated linear factors.
• When denominator contains repeated linear factors.
• When denominator contains non-repeated irreducible quadratic factors.
• Define definite integral.
• Define the fundamental theorem of integral calculus.
• Area under the curve.
• Find the area bounded by the curve.
• Find the area between the x-axis and the curve.
• Define the differential equations.
• Define the order of a differential equation.
• Define the degree of a differential equation.
• Define solution of a differential equation.
• Define general solution of a differential eqaution.
• Define particular solution of a differential equation.
• How many constants are involved in general solution of first-order differential equation.
• How many constants are involved in general solution of nth order.
• Solve the following differential equations.

# Chapter 4 (Introduction to Analytic Geometry) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 5
Type: Notes

In this chapter we will read these topics:

• What are the meanings of the word geometry?
• Who was the founder of geometry.
• What is Plane geometry.
• What is Solid geometry.
• What is analytic geometry?
• Define a coordinate plane and coordinate axes.
• Define abscissa and ordinate of a point.
• State and prove distance formula.
• Distance of point P(x,y) from x-axis is |y| and from y-axis is |x|.
• Distance measured non-parallel to coordinate axes is called undirected distance.
• Distance measured parallel to the coordinate axes is called directed distance.
• State and prove the Ration formula or Division formula.
• State and prove the Ration formula/Division formula for external division.
• Derive mid-point formula from ratio formula.
• Define Median of a triangle.
• What do you mean by concurrent lines.
• Define Centroid of a triangle.
• Show that medians of a triangle are concurrent.
• What are coordinates of centroid of a triangle.
• Define angle bisector and in-center of a triangle.
• Define the translation of axes.
• Write the equations of translation of axes.
• Define the rotation of axes.
• Write the equation of axes.
• Write the equation of rotation of axes.
• Define the inclination of a line.
• Define the slope or gradient of a line.
• Write formula for slope of a line passing through two points.
• What is the condition of two parallel lines.
• What is the condition of two perpendicular lines.
• When three points are said to be colinear.
• What is the equation of line parallel to x-axis.
• What is the equation of line parallel to y-axis.
• What do you mean by intercepts.
• State and prove the equation of line in slop-intercept form.
• Types of lines.
• State and prove equation of line in point-slope form.
• Derive the symmetric form of a line.
• State and prove equation of line in two-points form.
• Equation of non-vertical straight line passing through two points.
• State and prove equation of line in intercept form.
• State and prove equation of line in normal form.
• Describe the position of a point with respect to line.
• Point of intersection and condition of concurrent.
• Prove that altitudes of a triangle are concurrent.
• Prove that the right bisector of triangle are concurrent.
• Angle between two lines.
• Two lines l1 and l2 are parallel if and only if m1=m2.
• Find point of intersection of the lines.
• Find equation of line.
• Find acute angle.
• Find interior angles of triangles.
• Find interior angle of the quadrilateral.
• Find the center of the circumcircle of the triangle.
• Define the homogeneous equation of degree n in x and y variables.
• Define general homogeneous equation of second degree in x and y variables.
• Find a joint equation of the lines through the origin and perpendicular to the lines.
• Find the area of the bounded region.
• Area of tapezium

# Chapter 7 (Vectors) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 5
Type: Notes

In this chapter we will read these topics:

• Define scalar quantity.
• Define the vector quantity.
• Give the geometrical interpretation of a vector quantity.
• Define the magnitude of a vector.
• Define unit vector.
• Define null vector.
• Define negative of a vector.
• Define the multiplication of a vector by a scalar.
• Define equal vectors.
• Define parallel vectors.
• Define triangular law of the addition of vectos.
• What is the parallelogram law of vector addition?
• Describe the subtraction of two vectors.
• Define the position vector.
• Describe vector in a plane.
• Discuss the addition and subtraction of two vectors in a plane.
• Define the magnitude of a vector in R2.
• State and prove ratio formula.
• Introduction of a vector in space.
• Discuss the addition and subtraction of two vectors in space.
• Find the magnitude, length, a norm of a vector in space.
• Properties of a vector in space.
• State and prove a commutative property of addition of vector in space.
• State and prove the associative property of addition of vectors in space.
• State and prove the additive inverse of a vector in space.
• State and prove the distributive property of a vector in space.
• State and prove scalar multiplication property of a vector in space.
• Write the form of the vector in space.
• State and prove distance formula.
• Define direction angles and direction cosines of a vector.
• Scalar product of two vectors.
• Define scalar and dot product.
• Define the scalar product of vectors in a plane.
• Define the scalar product of vectors in a space.
• Define perpendicular vectors.
• Show that the zero vector is perpendicular to every vector.
• Properties of dot product.
• Define the angle between two vectors.
• Projection of a vector upon another.
• Cross product or vector product.
• Define the cross product of two vectors.
• Define the right-hand rule.
• Important results of the cross product.
• The analytical expression of cross products.
• Properties of cross product.
• Define the area of a parallelogram.
• Define area of a triangle.
• Scalar triple product of vectors.
• Define the scalar triple product of three vectors.
• Define parallelepiped.
• Define tetrahedron.
• Give the geometrical interpretation of the scalar triple product.
• Define the volume of a tetrahedron.
• Properties of scalar triple product.
• Define coplanar lines.
• Find the volume of a tetrahedron.
• Define words done by a force.
• Define moment of arm.

# Chapter 6 (Conic Section) FSc Part 2 Math

Level: F.Sc (Pre-Engg)
Class: 2nd Year
Board: Punjab Textbook Board
Subject: Mathematics
Total Exercises: 9
Type: Notes

In this chapter we will read these topics:

• Define cone, vertex of cone and axis of cone.
• Define nappes of cone and types of cone.
• Define Conic or Conic section.
• Define Circle.
• Define Point Circle.
• Define Ellipse.
• Define Parabola.
• Define Hyperbola.
• What is the difference between the diameter and chord of a circle?
• Derive the standard equation of circle.
• Find the equation of circle having center at origin and radius r.
• Find the parametric equation of circle having radius r and center at origin.
• When two circles touch each other externally.
• When two circles touch each other internally.
• Tangent to a circle is perpendicular to radius.
• Tangents and normals.
• Define tangent and normal to a curve.
• Find equation of tangent and normal to circle.
• Normal line is perpendicular to the line.
• Condition of tangency.
• Tangential distance.
• Equation of chord of contact.
• Analytic proofs of important properties of circle.
• Perpendicular dropped from the center of a circle on a chord bisects the chord.
• The perpendicular bisector of any chord of a circle passes through the center of the circle.
• The line joining the center of a circle to the mid-point of a chord is perpendicular to the chord.
• The slope of chord.
• Congruent chords of a circle are equidistance from center.
• Show that the measure of the central angle of minor arc is double the measure of the angle subtended in the corresponding major arc.
• An angle in a semi-circle is a right angle.
• The tangent to a circle at any point of the circle is perpendicular to the radial segment at that point.
• Perpendicular at outer end of a radial segment is tangent to circle.
• Define parabola.
• Derive the standard equation of parabola y2=4ax.
• Derive the general equation of a parabola.
• Write the relationship between the second-degree equation and parabola.
• Define focus, directrix, axis and vertex of parabola.
• Define tangent at vertex and chord of parabola.
• Define the focal chord and latus rectum of the parabola.
• What do you mean by the eccentricity of a parabola?
• Prove that length of latus rectum of a parabola is 4a.
• Point of a parabola which is closest to the focus is the vertex of parabola.
• Length of latus rectum.
• Define Reflecting property of parabola.
• Define ellipse.
• The fixed point is called the focus.
• Define the diameters of an ellipse.
• Define major axis of an ellipse and length of major axis.
• Define minor axis of an ellipse and length of minor axis.
• Define directrices of an ellipse.
• Define the Foci of an ellipse.
• Define vertices of an ellipse.
• Define co-vertices of an ellipse.
• Define chord of an ellipse.
• Define focal chord of an ellipse.
• Define latus rectum and length of latus rectum of an ellipse.
• Define center of an ellipse.
• Define semi-major axis and semi-minor axis of an ellipse.
• Write the parametric equations of an ellipse.
• Derive the equation of directrix of an ellipse.
• Define hyperbola.
• Define center of Hyperbola.
• Define focal axis or transvers axis of Hyperbola.
• Define the conjugate axis of Hyperbola.
• Define the vertices of Hyperbola.
• What is length of latus rectum of Hyperbola?
• Write the parametric equation of Hyperbola.
• Which conics re called conics.
• Discuss the sketch graph of the equation.
• Tangents and normals.
• Condition of tangency.
• Find the point of intersection ellipse and hyperbola.
• Find the point of intersection of conics.
• Define the translation of axes.
• Write the equation of the translation of axes.
• Define the rotation of axes.
• Write the equation of rotation of axes.
• Transformed equation.
• Define the general equation of second degree.
• When the general equation of second degree represents a circle.
• When the general equation of second degree represents a parabola.
• When the general equation of second degree represents a ellipse.
• When the general equation of second degree represents a hyperbola.
• Classification of conics by discriminant.
• Discuss the nature of conics.
• What is degenerate conic?
• Elements of the ellipse.
• Equation of major axis.
• Equation of minor axis.
• Equation of transformation.
• Elements of the hyperbola.
• Equation of focal axis.
• Equation of the conjugate axis.