CSET 3 REVIEW
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Calculus (22 Multiple Choice Items, 3 Constructed Response Items)
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Trigonometry
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a. Prove that the Pythagorean Theorem is equivalent to the trigonometric identity and that \
this identity leads to and .
Lesson 1. Proof of Pythagorean Theorems: https://www.youtube.com/watch?v=iEKT-yAuRgQ; https://www.youtube.com/watch?v=MyO2MFJbfi4
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b. Prove and apply the sine, cosine, and tangent sum formulas for all real values.
Lesson 2. Proof of Sin Sum Formula: https://www.youtube.com/watch?v=R0EQg9vgbQw
Lesson 3. Visual Proof of Sine and Cos Sum Formulas: https://www.youtube.com/watch?v=ea6-ctMYn_Q
Lesson 4. Proof of Cos Sum Formula: https://www.youtube.com/watch?v=gDOGT6NcD60
Lesson 5. Proof of Tan Sum Formula: https://www.youtube.com/watch?v=1qCXck4ZIrk
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c. Analyze properties of trigonometric functions in a variety of ways (e.g., graphing and solving problems, using the unit circle)
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Lesson 6. Properties of Trig Functions: https://www.youtube.com/watch?v=ypenWNeRdd4; https://www.youtube.com/watch?v=r26NNufSFYg; https://www.youtube.com/watch?v=soIt2TwV6Xk; https://www.youtube.com/watch?v=8Z60_yXX4xA; https://www.youtube.com/watch?v=ztlVm6COcK4; https://www.youtube.com/watch?v=Dis8_Ih7slo; https://www.youtube.com/watch?v=ZffZvSH285c
Lesson 7. Evaluating Trig Functions: https://www.youtube.com/watch?v=8DdyCeTEGOo; https://www.youtube.com/watch?v=QukM0eBDwNE; https://www.youtube.com/watch?v=9OdJ54fGgVs
Lesson 8. Symmetry of Trig Values: https://www.youtube.com/watch?v=tzQ7arA917E
Lesson 9. Use Trig Identities to Evaluate Expressions: https://www.youtube.com/watch?v=sGDbKmWmTDw
d. Apply the definitions and properties of inverse trigonometric functions (i.e., arcsin, arccos, and arctan)
Lesson 10. Introduction: https://www.youtube.com/watch?v=bBBUMHe900U; https://www.youtube.com/watch?v=JGU74wbZMLg; https://www.youtube.com/watch?v=hxjmtDXXCzU; https://www.youtube.com/watch?v=Idxeo49szW0
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Lesson 11. Evaluating Inverse Trig Functions: https://www.youtube.com/watch?v=aRVWs1tDarI; https://www.youtube.com/watch?v=m5DZQDzsJsE; https://www.youtube.com/watch?v=g9S4u8eQiww
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Lesson 12. Simplifying Expressions with Inverse Trig Functions: https://www.youtube.com/watch?v=Zud3aCeSLRs
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Lesson 13. Solving Trigonometric Equations: https://www.youtube.com/watch?v=IE0FxGegdMg; https://www.youtube.com/watch?v=ZkK4ifsQoGk; https://www.youtube.com/watch?v=iZihVtFaAko; https://www.youtube.com/watch?v=kEcbxiLeGTc; https://www.youtube.com/watch?v=YESk2q8QEOY; https://www.youtube.com/watch?v=CrayigBVBZo; https://www.youtube.com/watch?v=IE0FxGegdMg&list=RDQMDgpikCqDgLY; https://www.youtube.com/watch?v=fWI0frnVHy0; https://www.youtube.com/watch?v=7Eo-fuy0f7g; https://www.youtube.com/watch?v=_PPj7eeT_7E
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e. Apply polar representations of complex numbers (e.g., DeMoivre's Theorem)
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Lesson 14. Polar Representations of CI: https://www.youtube.com/watch?v=fbqakDV-IXc; https://www.youtube.com/watch?v=6HIlT6oSvXc; https://www.youtube.com/watch?v=8RasCV_Lggg; https://www.youtube.com/watch?v=hdlkr0t2R88; https://www.youtube.com/watch?v=u3Q_0WxxdeM
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Lesson 15. De Moivre's Theorem: https://www.youtube.com/watch?v=kEf9gt3umnU; https://www.youtube.com/watch?v=yf22lfJ1JCU; https://www.youtube.com/watch?v=y2voZK3-CHA; https://www.youtube.com/watch?v=1Eu66E_MoQ8; https://www.youtube.com/watch?v=irwItP7PiQg
f. Model periodic phenomena with periodic functions
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Lesson 16. Modeling: https://www.youtube.com/watch?v=kXKg56CPnpE; https://www.youtube.com/watch?v=RX0DY9eRp8g; https://www.youtube.com/watch?v=s4cLM0l1gd4; https://www.youtube.com/watch?v=6oeBPCzaugI; https://www.youtube.com/watch?v=DP6Drp69fGM; https://www.youtube.com/watch?v=Ogcd_Rhk0bU
g. Recognize equivalent identities, including applications of the half-angle and double-angle formulas for sines and cosines
Lesson 17. Examples: https://www.youtube.com/watch?v=OJz-fOzFbEc; https://www.youtube.com/watch?v=NK4MlZXepYQ; https://www.youtube.com/watch?v=IE8q4WRubC4; https://www.youtube.com/watch?v=d_NMpLSGpkU; https://www.youtube.com/watch?v=A1iKe4HkoXc; https://www.youtube.com/watch?v=uFbbF-IYFjM; https://www.youtube.com/watch?v=qWJ4HwJ_5LY; https://www.youtube.com/watch?v=Wh7hM_FsN3Y; https://www.youtube.com/watch?v=TjWa00qkYpI
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Assessment: Practice Problems Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7
CSET 3 Math Questions Do Problems 1-5 and 25
Limits and Continuity
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a. Derive basic properties of limits and continuity, including the Sum, Difference, Product, Constant Multiple, and Quotient Rules, using the formal definition of a limit
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Lesson 18. Definition of a Limit of a Function: https://www.youtube.com/watch?v=-ejyeII0i5c; https://www.youtube.com/watch?v=G0ax2x2_Em0; https://www.youtube.com/watch?v=Fdu5-aNJTzU; https://www.youtube.com/watch?v=0sCttufU-jQ; https://www.youtube.com/watch?v=dXr6aoJ1nVI
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Lesson 19. Properties of Limits: https://www.youtube.com/watch?v=lSwsAFgWqR8; https://www.youtube.com/watch?v=6webTCd5gEQ; https://www.youtube.com/watch?v=yzwtrqsvCvI; https://www.youtube.com/watch?v=GGQngIp0YGI; https://www.youtube.com/watch?v=5T8ifCP6eIg; https://www.youtube.com/watch?v=v_Nz6UUQ4HQ; https://www.youtube.com/watch?v=v_Nz6UUQ4HQ
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Lesson 20. Infinite Limits: https://www.youtube.com/watch?v=-vwcLvb9A0s; https://www.youtube.com/watch?v=xvFqomOpLrs; https://www.youtube.com/watch?v=tFHALKP22ao
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Lesson 21. Limits at Infinity: https://www.youtube.com/watch?v=kae8X6aplf0; https://www.youtube.com/watch?v=FVJNuukADeQ; https://www.youtube.com/watch?v=75xO9xy7TTQ; https://www.youtube.com/watch?v=mrKg_5CsfX0
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b. Show that a polynomial function is continuous at a point
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Lesson 22. Definition of a Continuous Function: https://www.youtube.com/watch?v=InDHwh1CvOg; https://www.youtube.com/watch?v=kdEQGfeC0SE; https://www.youtube.com/watch?v=oUgDaEwMbiU'; https://www.youtube.com/watch?v=NVQBZWTKygU; https://www.youtube.com/watch?v=SoMl89ZYJIg; https://www.youtube.com/watch?v=OEE5-M4aY4k
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Lesson 23. Polynomial function and continuity at a point: https://www.youtube.com/watch?v=OEE5-M4aY4k
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c. Apply the intermediate value theorem, using the geometric implications of continuity
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Lesson 24. Intermediate Value Theorem and Examples: https://www.youtube.com/watch?v=9xgO-EJ3sr0; https://www.youtube.com/watch?v=9wEHwFrUyOU; https://www.youtube.com/watch?v=TrQTK-B4bzM; https://www.youtube.com/watch?v=Rpug_8nTqyw
Assessment: Practice Problems Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6
CSET 3 Math Questions Do Problems 6-9
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Derivatives and Applications
a. Derive the rules of differentiation for polynomial, trigonometric, and logarithmic functions using the formal definition of derivative
Lesson 25. Definition of Derivative of a Function: https://www.youtube.com/watch?v=Vv1BUCkgsr8; https://www.youtube.com/watch?v=vzDYOHETFlo
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Lesson 26. Differentiation Rules of Algebraic Functions: https://www.youtube.com/watch?v=B2qJURxVyps; https://www.youtube.com/watch?v=v8AmvKHJqTU; https://www.youtube.com/watch?v=hCLfogkqzEk
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Lesson 27. Differentiation Rules for Transcendental Functions: https://www.youtube.com/watch?v=TDHI-aieyfk; https://www.youtube.com/watch?v=OjnOgoEu6CM; https://www.youtube.com/watch?v=0YH8BrlVTqk; https://www.youtube.com/watch?v=dylEXVIasqs
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Lesson 28. Proofs: https://www.youtube.com/watch?v=JjOndio6-g4; https://www.youtube.com/watch?v=c_CpxQdMejE; https://www.youtube.com/watch?v=ho87DN9wO70; https://www.youtube.com/watch?v=FztF4sJ66rY
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Lesson 29. Implicit Differentiation: https://www.youtube.com/watch?v=kk2HilVYAPE; https://www.youtube.com/watch?v=5yTVUZCaU6k; https://www.youtube.com/watch?v=sL6MC-lKOrw; https://www.youtube.com/watch?v=_kLWcuJzYh8; https://www.youtube.com/watch?v=2dv_PfEFZXY
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b. Interpret the concept of derivative geometrically, numerically, and analytically (i.e., slope of the tangent, limit of difference quotients, extrema, Newton's method, and instantaneous rate of change)
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Lesson 30. Graphing using derivatives: https://www.youtube.com/watch?v=hIgnece9ins; https://www.youtube.com/watch?v=4aYGmexcKnc
Lesson 31. Tangents: https://www.youtube.com/watch?v=1KwW1v__T_0; https://www.youtube.com/watch?v=GH8-URjRQpQ; https://www.youtube.com/watch?v=7EFYoQ6H7Tw
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Lesson 32. Newton's Method: https://www.youtube.com/watch?v=1uN8cBGVpfs; https://www.youtube.com/watch?v=HaUKd-UXfMQ; https://www.youtube.com/watch?v=xdLgTDlFwrc
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Lesson 33. Instantaneous rate of change: https://www.youtube.com/watch?v=hSK7f4durxI; https://www.youtube.com/watch?v=MRDPyXgxN78; https://www.youtube.com/watch?v=jlihNi_Mkos; https://www.youtube.com/watch?v=4Up5gsDeluw; https://www.youtube.com/watch?v=dPZcJSctp38
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c. Interpret both continuous and differentiable functions geometrically and analytically and apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule
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Lesson 34. Rolle's Theorem and MVT: https://www.youtube.com/watch?v=0jiVRRKmGoE; https://www.youtube.com/watch?v=jgDykHxQxqM; https://www.youtube.com/watch?v=Du3PuJjME8k; https://www.youtube.com/watch?v=xYOrYLq3fE0; https://www.youtube.com/watch?v=M2b2ok-jto4
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Lesson 35. L'Hopital's Rule: https://www.youtube.com/watch?v=PdSzruR5OeE; https://www.youtube.com/watch?v=RQSnbJd04GY; https://www.youtube.com/watch?v=thEnl_gCjXA; https://www.youtube.com/watch?v=Haapl1SrB5I; https://www.youtube.com/watch?v=Zd7wd24jeok; https://www.youtube.com/watch?v=6g_g9-AU188; https://www.youtube.com/watch?v=BU2QhCk3ExY
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d. Use the derivative to solve rectilinear motion, related rate, and optimization problems
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Lesson 36. Optimization: https://www.youtube.com/watch?v=lzLgtk-lrW0; https://www.youtube.com/watch?v=Zq7g1nc2MJ8; https://www.youtube.com/watch?v=mamH094uw_U; https://www.youtube.com/watch?v=EOJbmMB8uCQ
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Lesson 37. Related Rates: https://www.youtube.com/watch?v=BmWmue6sDFg; https://www.youtube.com/watch?v=_kbd6troMgA; https://www.youtube.com/watch?v=kQF9pOqmS0U; https://www.youtube.com/watch?v=kBVDSu7v8os
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Lesson 38. Rectilinear Motion: https://www.youtube.com/watch?v=9ot21Up_mj4; https://www.youtube.com/watch?v=HDntI7zfBNs; https://www.youtube.com/watch?v=HzN99FYCg5I; https://www.youtube.com/watch?v=pFeuGMMiZWw
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e. Use the derivative to analyze functions and planar curves (e.g., maxima, minima, inflection points, concavity)
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Lesson 39. Graphing Functions Using Derivative Concepts: https://www.youtube.com/watch?v=15awMHeP1Yc; https://www.youtube.com/watch?v=pQT2fZRMcf4; https://www.youtube.com/watch?v=3TJLOCYrTes; https://www.youtube.com/watch?v=lDY9JcFaRd4; https://www.youtube.com/watch?v=RoxefQ_Qgm8; https://www.youtube.com/watch?v=kTWXVJHdrwM; https://www.youtube.com/watch?v=Gy9jKk28XHY
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Assessment: Practice Problems Problem 1 Problem 2 Problem 3 Problem 4 Problem 5
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CSET 3 Math Questions Do Problems 10-15
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Integrals and Applications
a. Derive definite integrals of standard algebraic functions using the formal definition of integral and interpret the concept of a definite integral geometrically, numerically, and analytically (e.g., limit of Riemann sums)
Lesson 40. Finding Area using Reimann Sums: https://www.youtube.com/watch?v=AkUa9Fkz2rw; https://www.youtube.com/watch?v=gFpHHTxsDkI; https://www.youtube.com/watch?v=ODwkTt0RMDg; https://www.youtube.com/watch?v=Himr2l8Rd18
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Lesson 41. Definite Integral: https://www.youtube.com/watch?v=xR4AnXDBnsw; https://www.youtube.com/watch?v=rCWOdfQ3cwQ; https://www.youtube.com/watch?v=0dDIPzqKgYk; https://www.youtube.com/watch?v=hTTDI49LhjE; https://www.youtube.com/watch?v=8BzIOowS77A; https://www.youtube.com/watch?v=nopnFjMy3rQ
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Lesson 42. Properties of Definite Integrals: https://www.youtube.com/watch?v=MM0FTzvedH4; https://www.youtube.com/watch?v=wycadSRDID4
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c. Prove the fundamental theorem of calculus, and use it to interpret definite integrals as antiderivatives
Lesson 43. FTC: https://www.youtube.com/watch?v=C7ducZoLKgw; https://www.youtube.com/watch?v=PGmVvIglZx8
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Lesson 44: Proof of FTC: https://www.youtube.com/watch?v=pWtt0AvU0KA; https://www.youtube.com/watch?v=mhb0epc6aFk
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Lesson 45: Applications of FTC: https://www.youtube.com/watch?v=FcLeaD3UII4; https://www.youtube.com/watch?v=TQTDkpZP02A; https://www.youtube.com/watch?v=lvdnTXZGEF0&list=PL823769CE78CC2EE4
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d. Apply the concept of integrals to compute the length of curves and the areas and volumes of geometric figures
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Lesson 46: Find length of a curve: https://www.youtube.com/watch?v=neOQXrZv1rY; https://www.youtube.com/watch?v=PwmCZAWeRNE; https://www.youtube.com/watch?v=OhISsmqv4_8; https://www.youtube.com/watch?v=qFowh4Ir7GU; https://www.youtube.com/watch?v=DNDAwWIL5FY
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Lesson 47: Area Problems: https://www.youtube.com/watch?v=4bZyfvKazzQ; https://www.youtube.com/watch?v=RebqJ1wqP2I; https://www.youtube.com/watch?v=70NQ3ISYihw; https://www.youtube.com/watch?v=I3bxriE3XbM; https://www.youtube.com/watch?v=B5441_DREY0; https://www.youtube.com/watch?v=LbTH7MGMNjk
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Lesson 48: Volume Problems: https://www.youtube.com/watch?v=E5OOMbz5jZk; https://www.youtube.com/watch?v=GJOJl47l2_4; https://www.youtube.com/watch?v=lBSLPUbYFsU; https://www.youtube.com/watch?v=nZqOKc067Z8; https://www.youtube.com/watch?v=R_aqSL-q6_8; https://www.youtube.com/watch?v=43AS7bPUORc; https://www.youtube.com/watch?v=V6nTsxumjgU; https://www.youtube.com/watch?v=cYJRMejnBqI; https://www.youtube.com/watch?v=puSlVA6mwNQ
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e. f. Solve separable first-order differential equations and apply them to growth and decay problems
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Lesson 49: Examples: https://www.youtube.com/watch?v=nNHlSB6b1HU; https://www.youtube.com/watch?v=6vUjGgI8Dso; https://www.youtube.com/watch?v=DL-ozRGDlkY; https://www.youtube.com/watch?v=XExEixAPK6s; https://www.youtube.com/watch?v=Et4Y41ZNyao; https://www.youtube.com/watch?v=C5-lz0hcqsE; https://www.youtube.com/watch?v=xVWCfMe97ws; https://www.youtube.com/watch?v=9wwaC_jnQQ0; https://www.youtube.com/watch?v=3jpiW_oueaA
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Assessment: Practice Problems Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6
CSET 3 Math Questions Do Problems 16-20 and 26
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Sequences and Series
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a. Derive and apply the formulas for the sums of finite arithmetic series and finite and infinite geometric series (e.g., express repeating decimals as a rational number)
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Lesson 50: Arithmetic Sum: https://www.youtube.com/watch?v=RM644gFKo_g; https://www.youtube.com/watch?v=Uy_L8tnihDM
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Lesson 51: Finite Geometric Series: https://www.youtube.com/watch?v=AXP5PGSaaYk; https://www.youtube.com/watch?v=i8THsl3AYFI
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Lesson 52: Infinite Geometric Series: https://www.youtube.com/watch?v=b-7kCymoUpg; https://www.youtube.com/watch?v=yYxzq_O18Mg; https://www.youtube.com/watch?v=x2Xmldq0ll8; https://www.youtube.com/watch?v=ocVBZjuJUdc
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Lesson 53: Expressing Repeating Decimals as Rational Numbers: https://www.youtube.com/watch?v=2BgWWsypzLA; https://www.youtube.com/watch?v=7tDK_UjdWOs
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b. Determine convergence of a given sequence or series using standard techniques (e.g., ratio, comparison, integral tests)
Lesson 54: Definitions and Examples: https://www.youtube.com/watch?v=FoNLQvf4NUs; https://www.youtube.com/watch?v=lfZGtjSWcQs; https://www.youtube.com/watch?v=muqyereWEh4; https://www.youtube.com/watch?v=NjDk8HiPOhk; https://www.youtube.com/watch?v=7_52IGDHrqY
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Lesson 55: Ratio Test - https://www.youtube.com/watch?v=iy8mhbZTY7g; https://www.youtube.com/watch?v=av947KCWf2U; https://www.youtube.com/watch?v=AwJ0P8B25tY; https://www.youtube.com/watch?v=rUis1mSzwyA
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Lesson 56: Root Test - https://www.youtube.com/watch?v=vDdDLfIya0I
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Lesson 57: Comparison Test - https://www.youtube.com/watch?v=b7OetOcd188; https://www.youtube.com/watch?v=0tXxFPHzFFI; https://www.youtube.com/watch?v=AtbZZiSLemQ; https://www.youtube.com/watch?v=xesQnFWw8f8
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Lesson 58: Integral Test - https://www.youtube.com/watch?v=xRyXz_UZ14Q; https://www.youtube.com/watch?v=8jPpNK4GIVs; https://www.youtube.com/watch?v=ojztrQMqLgE; https://www.youtube.com/watch?v=XHUxLLu_c_0; https://www.youtube.com/watch?v=wXDXe9YI4mc; https://www.youtube.com/watch?v=gvZ7UIaC50M
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Lesson 59: Divergence Test - https://www.youtube.com/watch?v=1yInzfzDDKY
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Lesson 60: Alternating Series Test - https://www.youtube.com/watch?v=91qVGeyTl44
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Lesson 61: P Series Test for C or D - https://www.youtube.com/watch?v=w9fC8RaklhA; https://www.youtube.com/watch?v=gvZ7UIaC50M
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Lesson 62: Strategies for Testing Series - https://www.youtube.com/watch?v=DvadVYHf3pM; https://www.youtube.com/watch?v=ac00oud9z4A
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c. Calculate Taylor series and Taylor polynomials of basic functions
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Lesson 63: Taylor and MacLaurin Series - https://www.youtube.com/watch?v=epgwuzzDHsQ&list=PLA5FB0434B1C097D0; https://www.youtube.com/watch?v=cjPoEZ0I5wQ; https://www.youtube.com/watch?v=3d6DsjIBzJ4; https://www.youtube.com/watch?v=Os8OtXFBLkY
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Lesson 64: Finding Taylor polynomials of basic functions - https://www.youtube.com/watch?v=epgwuzzDHsQ; https://www.youtube.com/watch?v=19x213y_uk4; https://www.youtube.com/watch?v=8SsC5st4LnI; https://www.youtube.com/watch?v=UINFWG0ErSA
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Assessment: Practice Problems Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7
CSET 3 Math Questions Do Problems 21-24