MATH 2300 Practice-Exam 1 (Spring 2007)

# MATH 2300 Practice-Exam 1 (Spring 2007) - MATH...

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Unformatted text preview: MATH 2300 (CALCULUS 2), SPRING 2007 7 1ST TEST PRACTICE EXAM 1 MATH 2300 (Calculus 2) Spring 2007 lst midterm PRACTICE exam The present is a practice exam. You should not assume that the problems of the actual exam you will take are analogous, similar, or in any way related to the problems here. In every other aspect, this practice exam is a realistic exam, with problems of a diﬁiculty comparable to the ones you will be asked to solve the day of the test. It is not intended as a set of review problems but its purpose is to allow you to time yourself with something like “the real thing.” If you use it for its intended purpose, keep in mind that the day of the test you will be under more pressure and that it may take you longer to ﬁnish your work. _ l 1. [?? pts_ Evaluate ﬂ dzr. ﬁ 2. [?? pts Evaluate /cos(2m) sin(4m) dm. 3. [?? pts Evaluate /tanm secgmdm. 4. [?? pts Evaluate \$2 / d“ 1 5. pts Evaluate/mdm. 2 6. [?? pts Evaluate/mdm. 7. [?? pts: Make appropriate substitutions to simplify the following integrals, but do not evaluate them! NOTE: Do “the ﬁrst” and “most obvious” simpliﬁcation. There may be more after that, but one is certainly “the ﬁrst.” 8. [?? pts] Evaluate the improper integral / (I: 2 mVIHm or show that it does not exist. MATH 2300 (FALL 2007) 7 TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS (BASIC FOR ULAS) 1. Addition Theorems sin(a: :: y) : sin 1: cos y :: cos a: sin y sinh(a7 :: y) : sinh a7 cosh y :: cosh a7 sinh y c0s(a: :: y) : cos 1: cos y :: sin a: sin y c0sh(a7 :: y) : cosh a7 cosh y :: sinh a7 sinh y tana7:tan t h :1; h tan(:v :: y) 2 7y tanh(a7 :: y) : w 1 7 tan 1? tan y 1 :: tanh a: tanh y Product decomposition 1 1 sina: cos y : 5 [sin(a: 7 y) + sin(a7 + 31)] sinha: cosh y : 5 [sinh(a7 + y) + sinh(a: 7 31)] 1 1 sing? sin y : 5 [cos(a: 7 y) 7 c0s(a7 7 31)] sinha: sinh y : 5 [cosh(a7 7 y) 7 cosh(a7 7 31)] 1 1 cos 1: cos y : 5 [cos(a: 7 y) + c0s(a7 7 31)] cosha: cosh y : 5 [cosh(a7 7 y) + cosh(a7 7 31)] Double-angle and Half-angle formulas sin 2:v = 2 sin :v cos :v sinh 2:v = 2 sinh :v cosh 30 cos 2:v = cos2 :v — sin2 30 cosh 2:v = cosh2 m + sinh2 :v : 2c0s2a771 : 2c0sh2ar71 : 172sin2a7 : 2sinh2a7+1 . 1 . 1 sin2 E = — (1 — cos x) sinh2 E = — (coshm — 1) 2 2 2 2 2 :v 1 2 m 1 cos — = — 1 cos x cosh — = — cosh m 1 2 2 ( + ) 2 2 ( + ) 2. Derivatives of Inverse Functions d sin’1 :v — 1 < 1 d sinh’1 :v — 1 d1? — 7172 d1? — V1 +332 (1 1 d 1 —cos_1:v = 7— <1 —cosh_1:v = 7 CE>1 d1? 7 172 d1? 1:2 7 1 d _1 1 d _1 1 —tan m = —tanh :v = :v 1 dz? 1 7 1:2 dz? 1 7 172 I I < d _1 1 d _1 1 — cot a7 : 7 —c0th a: : a? > 1 dz? 1 7 1:2 dz? 1 7 172 I I d sec’1 1 1 \> 1 d sech’1 1 0 < < 1 — I : — l7 — 17 : 7— 17 d1? bah/\$2 — 1 d1? \$\/1 — m2 d ,1 1 d ,1 1 —csc a: : 7— a? > 1 —csch a7 : 7— a: 0 0190 ‘m‘x/m271 I I dye “UM—1+1; 75 3. Logarithmic form of Inverse Hyperbolic Functions sinh—117 : ln(a:+\/a72+1) cosh—117 : ln(a:+\/a7271) 1 1 1 1 tanh_1m : —1n( +\$) Goth—1m = —ln(m+ ) 2 1 — an 2 m — 1 1 \/1 7 2 1 \/1 2 sech_1a: : 1n — + 717 (sch—117 : ln — + i 4. Integrals of Trigonometric Functions and Powers of Trigonometric Functions - - n 1 - n71 n _ 1 - n72 smr d:v = — cos x sm m dm 2 —— s1n :v cos x + sm m d:v n n - n 1 7171 ~ 77’ 7 1 71—2 cosa: d1: : Slnl? cos 1? d1? : — cos 17 \$1111? + cos 1: d1? n n n 1 n71 n72 tanm d:v = 1n \sec 30‘ tan :v d:v = 1 tan :v — tan m d:v n _ - n 1 17.71 17.72 cot 17 d1: : ln1sma71 cot 1: d1? : 7 1 cot a7 + cot 17 d1? n 7 TL 1 7172 n _ 2 71—2 seer d:v = 1n1secm + tan 30‘ sec 30 dm 2 1 sec 30 tanm + 1 sec 30 d:v n — n — n 1 n72 77’ 7 2 17.72 csca: d1: : ln1csc a7 7 cot 17‘ csc 17 d1? : 7 1 csc a7 cot a: + 1 csc 1: d1? n 7 n 7 ALL THE REST YOU HAVE TO BE ABLE TO WORK IT OUT FROM THESE EXPRESSIONS ...
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MATH 2300 Practice-Exam 1 (Spring 2007) - MATH...

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