Math 146 - Calc II
Review for Test 1
Test 1 will consist in 5 problems and 1 theoretical question (for extra credit). You are allowed a ½ page
with formulas on one side only, but with no examples or solutions. The page should be turned in,
together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other
homework problems, and review the proofs of the theorems.
6.1 Volume by slicing
- Integral of cross-section area
- Cavalieri’s principle (consequence)
- Solids of revolution (special description of solid => special cross-section):
1) Disk method (cross-sections = disks)
2) Washer method (c-s = washers)
6.2 Volumes by cylindrical shells
- Like “Volume by slicing”, except using cylindrical shells
- Integral of cross-section area (cylinders)
6.3 Lengths of plane curves
- Parametric curves, graph of a function:
- Differential formula: L= ds, ds=(x’2+y’2)½ dt or ds=(1+y’2)½ dt
6.4 Moments and the Center of Mass
- Center of Mass, Torque of a mass: mgx; moment: mx
- Wires and Thin rods; thin flat plate (1-integrals)
6.5 Areas of surfaces of revolution and the Theorem of Pappus
- Surface area:  L(x)ds, L(x)=2 r(x)
- Theorems of Pappus
1) Volume of Solid of revolution: Area(cross-section) x Length(Centre of Mass Circle)
2) Area of Surface of Revolution= Area(cross-section) x Length(Centre of Mass Circle)
6.6 Work
- Work done by a constant force, work done by a variable force in 1-dimension
- Elastic force & Hooks law for springs (F=-kx)
- Gravity & Pumping liquids from containers (F=mg)
6.7 Fluid Pressures and Forces
- Pressure = Force / unit of surface; the Pressure-Depth Equation: p=wh
- Constant-depth formula for the fluid force: F=pA
- Variable-depth formula: F=p dA, vertical plate: dA=width x dh
- Fluid forces and centroids: F=w hc A
Math 146 - Calc II
Review for Test 2
Test 1 will consist in 5 problems and 1 theoretical question (for extra credit). You are allowed a ½ page
with formulas on one side only, but with no examples or solutions. The page should be turned in,
together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other
homework problems, and review the proofs of the theorems.
7.1 The Logarithm define as an Integral
- Def. ln x=1/t dt, the number e, graph and range of ln x
- The derivative of ln x and of ln|x|, the integral of ln u (u<0 and u>0)
- The inverse of ln(x): the natural exponential (derivative, integral, laws of exponents)
- General exponential function a^x, log_a(x), derivatives and integral
7.2 Exponential Growth and Decay
- The law of exponential change; unlimited population growth
- Continuous compounded interest: k times/year and continuous limit
- Applications: radioactivity, heat transfer
7.3 Relative rates of growth
- Comparing growth rates: asymptotically lim f(x)/g(x)
- Order of magnitude and Oh-notation: f=o(g)  lim f/g=0, f=O(g)  lim f/g>0 or f/g<M for large x.
7.4 Hyperbolic functions
- Even and odd parts of the exponential
- Def. hyperbolic functions sinh, cosh, tanh, coth
- Main identity, geometric interpretation: circle & (equilateral) hyperbola
- Derivatives and integrals
8.1 Basic Integration Formulas
- Substitution rule
8.2 Integration by parts
- Product rule in integral form, integration by parts formula
- Evaluating definite integrals by parts
8.3 Integration of rational functions by partial fractions
- General description of the Method
- Heaviside cover-up method for linear factors
8.4 Trigonometric integrals
- Products of powers of sines and cosines, eliminating square roots
- Integrals of powers of tan x and sec x; products of sines and cosines
8.5 Trigonometric substitutions: three basic substitutions
8.6 Integral Tables and Computer Algebra Systems
- Integral tables; Reduction formulas; Non-elementary integrals; Integration with a CAS
8.7 Numerical Integration
- Trapezoidal approximations – the trapezoidal rule; error estimate for the trapezoidal rule
- Simpson’s rule: approximating using parabolas; eror estimate for Simpson’s rule
8.8 Improper integrals
- Infinite limits of integration, integrands with vertical asymptotes
- Tests for convergence and divergence: direct comparison test, limit comparison test
Math 146 - Calc II
Review for Test 3
Test 1 will consist in 5 problems and 1 theoretical question (for extra credit). You are allowed a ½ page
with formulas on one side only, but with no examples or solutions. The page should be turned in,
together with the exam.
Please review the theory first, and work the suggested problems. Then continue with the other
homework problems, and review the proofs of the theorems.
Ch. 9 Further applications of integration
- From integral (definite/indefinite) to more general DE
9.1 Slope fields and separable differential equations
- General 1st ODE
- Slope fields: viewing solution curves – use calculator
- Separate equations
- Torricelli’s law – Ex.5
9.2 First-order linear DE
- The growth/decay equation, having a linear form
- Standard form, Ex.1
- Solving linear equations
- RL Circuits
- Mixture problems
9.3 Euler’s Method
- Euler’ method
- Improved Euler’s method
9.4 Graphical solutions of autonomous DE
- Equilibrium values and phase lines
- Stable and unstable equilibria
- Population growth in a limited environment
9.5 Applications of 1st ODE
- Resistance proportional to velocity
- Modeling population growth
- relative growth rate
- carrying capacity
- Orthogonal trajectories
Ch. 11 Infinite sequences and series
11.1 Sequences
- Infinite sequences
- Convergence and divergence
- Diverges to infinity
- Calculating limits of sequences
- The sandwitch theorem for sequences
- Using l’Hospital Rule
- Commonly occurring limits
- Recursive definitions
- Bounded non-decreasing sequences
- The non-decreasing Theorem
11.2 Infinite series
- Geometric series
- Divergent series
- The nth-term test for divergence
- Combining series
- Adding or deleting terms
- Reindexing
11.3 The integral test
- Non-decreasing partial sums
- The integral test
11.4 Comparison tests
- The comparison test
- The limit comparison test
11.5 The ratio and root test
- The ratio test
- The root test
11.6 Alternating series, absolute and conditional convergence
- Alternating series, harmonic series
- The Alternating Series Estimation Theorem
- Absolute and conditional convergence
- Absolute convergent
- Conditionally convergent
- The absolute convergence test
- Rearranging series
Math 146 - Calc II : Review for Final Exam
Final Exam will consist in 10 problems. You are allowed one page with formulas, but with no examples
or explicit solutions. The page should be turned in, together with the exam.
Please review the theory first, and work the suggested problems. Continue working the other homework
problems, and review the proofs of the theorems.
For chapters 6-9, 11-12 see the previous Reviews for Tests 1,2,3.
Calculator issues:
- Intersection points (f(x)=g(x)), zeros, derivatives, integration, graphing;
- Plotting slope fields and solutions to 1st order DE (adjusting the Window: Xmax, Ymax);
- Investigating limits, sequences, and series; Taylor series.
11.7 Power series
- Power series and convergence
- The convergence theorem for power series
- The radius of convergence of a power series
- Term-by-term differentiation
- Term-by-term differentiation theorem
- Term-by-term integration theorem
- Multiplication of power series
11.8 Taylor and MacLaurin series
- Series representations
- Taylor and MacLaurin Series
- Taylor polynomials
- Taylor polynomial of order n
11.9 Convergence of Taylor Series; error estimates
- Taylor Theorem / Taylor’s formula
- Estimating the remainder
- Truncation error
- Combining Taylor series
- Euler’s identity
11.10 Applications of power series
- The binomial series for powers and roots
- Power series solutions of DE
- Evaluating nonelementary integrals
- Arctangents
- Evaluating indeterminate forms
11.11 Fourier Series
- Fourier series, finding coefficients
- Convergence of Fourier Series