Solution to Review Problems for Midterm II

Size: px
Start display at page:

Download "Solution to Review Problems for Midterm II"

Transcription

1 Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions (a) f(x) 2x + 3 (b) f(x) 1 2x+3 Solution: (a) First, let us find te expression f(x + ) f(x) 2(x + ) + 3 2x + 3 2x x + 3 Now f(x+) f(x) 2x+2+3 2x+3 2x+2+3 2x+3 2x x+3 2x x+3 ( 2x+2+3) 2 ( 2x+3) 2 ( 2x x+3) 2x+2+3 (2x+3) ( 2x x+3) Tus f (x) lim 0 f(x+) f(x) lim 0 2 2x x+3 2 2x+3+ 2x x+3 1 2x+3 (b) First, let us find te expression 1 f(x + ) f(x) (x+)+3 2x+3 2x ( 2x x+3) ( 2x x+3) 2x+3 2x+3 2x+2+3 2x+3 (2x+2+3) 2x+3 2x 2 3 (2x+2+3)(2x+3) (2x+2+3)(2x+3) (2x+2+3)(2x+3) (2x+2+3)(2x+3) Now f(x+) f(x) 2) (2x+2+3)(2x+3) 2 (2x+2+3)(2x+3) 2 f (x) lim 0 f(x+) f(x) lim (2x+3)(2x+3) 2 (2x+3) 2 (2x+2+3)(2x+3) (2x+2+3)(2x+3) 2 (2x+2+3)(2x+3) Tus 2 Find te derivative of te following functions and simplify your answers (a) 12x 3 + 4x 2 x 2 (b) (1 + 4x)e 4x (c) (sec(x) + tan(x)) 3 (d) x cos(x) 6x sin(x) 6 cos(x) (e) ( cos(x) 1+sin(x) ) Solution: (a) First, we rewrite 12x 3 x 2 + 4x 2 12x 3 x 2 + 4x 2 (12x 3 +4x 2 x 2 ) (12x 3 x 2 +4x 2 60x x 3 8x ) 12 x 4 3 ( 2)x 3 +4 ( 2 So ) x (b) Applying te product rule, we ave ((1 + 4x)e 4x ) (1 + 4x) e 4x + (1 + 4x)(e 4x ) 4e 4x + (1 + 4x)e 4x ( 4x) 4e 4x + (1 + 4x)e 4x ( 4) 4e 4x 4e 4x 16xe 4x 16xe 4x (c) [(sec(x) + tan(x)) 3 ] 3(sec(x) + tan(x)) 2 (sec(x) + tan(x)) 3(sec(x) + tan(x)) 2 (sec(x) tan(x) + sec 2 (x)) 3(sec(x) + tan(x)) 2 sec(x) (sec(x) + sec(x)) 3 sec(x)(sec(x) + tan(x)) 3 (d)(x cos(x) 6x sin(x) 6 cos(x)) (x cos(x)) (6x sin(x)) (6 cos(x)) x 4 cos(x) + x ( sin(x)) 6 sin(x) 6x cos(x) 6( sin(x)) x 4 cos(x) x sin(x) 6 sin(x) 6x cos(x) + 6 sin(x) x 4 cos(x) x sin(x) 6x cos(x) MATH 180: page 1 of 6

2 Solution to Review Problems for Midterm II MATH 180: page 2 of 6 (e) [( cos(x) 1+sin(x) ) ] ( cos(x) 1+sin(x) )4 ( cos(x) 1+sin(x) ) ( cos(x) ( cos(x) 1+sin(x) )4 ( ( sin(x))(1+sin(x)) cos(x) cos(x) (1+sin(x)) 2 ( cos(x) 1+sin(x) )4 ( sin(x) 1 (1+sin(x)) 2 ) ( cos(x) 1+sin(x) )4 ( (cos(x)) (1+sin(x)) cos(x)(1+sin(x)) ) (1+sin(x)) 2 ) ( cos(x) 1+sin(x) )4 ( sin(x) sin2 (x) cos 2 (x) ) (1+sin(x)) 2 1+sin(x) )4 1 ( ) cos4 (x) (1+sin(x)) (1+sin(x)) 3 Find te derivative of te following functions You don t ave to simplify your answer (a) (2x + 1) 3 (1 + e 2x ) (b) (2x+1)3 (c) tan(sin(xe x )) (d) cot 6 ( 2) (1+e 2x ) t (e) (f) e 4x sec(x2) (g) sin 3 (2t) cos 3 (2t) () x 3 tan 3 ((1 + x 2 ) 2 ) 2 +e x2 (i) ex2 csc(3x) x 2 (j) x 4 e 3x cos(x) (k) sin (2x) (1+x 2 ) 2 x Solution:(a) [(2x + 1) 3 (1 + e 2x ) ] {}}{}{{} [(2x + 1) 3 ] }{{} (1 + e2x ) + (2x + 1) 3 [(1 + e 2x ) ] product rule x cos3 (2x) 3 (l) 1 + t cos(t 2 ) 2t3 3 sin(t2 ) {}}{ 3(2x + 1) 2 (2x + 1) }{{} (1 + e2x ) + (2x + 1) 3 (1 + e 2x ) 4 (1 + e 2x ) 3(2x + 1) 2 2 (1 + e 2x ) + (2x + 1) 3 (1 + e 2x ) 4 e 2x (2x) 6(2x + 1) 2 (1 + e 2x ) + (2x + 1) 3 (1 + e 2x ) 4 e 2x 2 6(2x + 1) 2 (1 + e 2x ) + 10(2x + 1) 3 (1 + e 2x ) 4 (b) First, we can rewrite (2x+1)3 (2x + 1) 3 (1 + e 2x ) (1+e 2x ) [ (2x+1)3 ] [(2x + 1) 3 (1 + e 2x ) ] (1+e 2x ) {}}{}{{} [(2x + 1) 3 ] }{{} (1 + e2x ) + (2x + 1) 3 [(1 + e 2x ) ] product rule {}}{ 3(2x + 1) 2 (2x + 1) }{{} (1 + e2x ) + (2x + 1) 3 ( ) (1 + e 2x ) 6 (1 + e 2x ) 3(2x + 1) 2 2 (1 + e 2x ) + (2x + 1) 3 ( ) (1 + e 2x ) 6 e 2x (2x) 6(2x + 1) 2 (1 + e 2x ) + (2x + 1) 3 ( ) (1 + e 2x ) 6 e 2x 2 6(2x + 1) 2 (1 + e 2x ) 10(2x + 1) 3 (1 + e 2x ) 6 (c) [tan(sin(xe x ))] sec 2 (sin(xe x ))[sin(xe x )] sec 2 (sin(xe x )) cos(xe x )(xe x ) sec 2 (sin(xe x )) cos(xe x )(e x + xe x ) (d) Note tat cot 6 ( 2 t ) (cot(2t 1 )) 6 So [cot 6 ( 2 t )] [(cot(2t 1 )) 6 ] 6(cot(2t 1 )) [cot(2t 1 )] 6(cot(2t 1 )) [ csc 2 (2t 1 )(2t 1 ) 6(cot(2t 1 )) [ csc 2 (2t 1 )]( 2t 2 ) (e) We can rewrite 4 x 2 +e x2 (x 2 +e x2 ) 1 4 [(x 2 + e x2 ) 1 4 ] [(x 2 + e x2 ) 1 4 ] (x 2 + e x2 ) 1 4 So ( 4x 2 +e x2 )

3 MATH 180: page 3 of 6 Solution to Review Problems for Midterm II ( 1 4 ) (x2 + e x2 ) 4 (x 2 + e x2 ) ( 1 4 ) (x2 + e x2 ) 4 (2x + e x2 (x 2 ) ) ( 1) 4 (x2 + e x2 ) 4 (2x + e x2 (2x)) (f) (e sec(x2) ) e sec(x2) (sec(x 2 )) }{{} esec(x2) sec(x 2 ) tan(x 2 )(x 2 ) }{{} e sec(x2) sec(x 2 ) tan(x 2 ) (2x) (g) (sin 3 (2t) cos 3 (2t)) (sin 3 (2t)) cos 3 (2t) + sin 3 (2t)(cos 3 (2t)) 3(sin 2 (2t)) (sin(2t)) cos 3 (2t) + sin 3 (2t) 3 (cos 2 (2t))(cos(2t)) 3(sin 2 (2t)) cos(2t) 2 cos 3 (2t) + sin 3 (2t) 3 (cos 2 (2t))( sin(2t)) 2 () [x 3 tan 3 ((1 + x 2 ) 2 )] (x 3 ) tan 3 ((1 + x 2 ) 2 ) + x 3 [tan 3 ((1 + x 2 ) 2 )] 3x 2 tan 3 ((1 + x 2 ) 2 ) + x 3 3 tan 2 ((1 + x 2 ) 2 )[tan((1 + x 2 ) 2 )] 3x 2 tan 3 ((1 + x 2 ) 2 ) + x 3 3 tan 2 ((1 + x 2 ) 2 )[sec 2 ((1 + x 2 ) 2 )][(1 + x 2 ) 2 ] 3x 2 tan 3 ((1 + x 2 ) 2 ) + x 3 3 tan 2 ((1 + x 2 ) 2 )[sec 2 ((1 + x 2 ) 2 )] 2 (1 + x 2 ) (2x) (i)note tat ex2 csc(3x) x 2 (1+x 2 ) 2 (e x2 csc(3x) x 2 )(1 + x 2 ) 2 So ( ex2 csc(3x) x 2 ) [(e x2 csc(3x) x 2 )(1 + x 2 ) 2 ] (1+x 2 ) 2 [(e x2 csc(3x) x 2 )] (1 + x 2 ) 2 + (e x2 csc(3x) x 2 )[(1 + x 2 ) 2 ] ([e x2 csc(3x)] 2x)(1 + x 2 ) 2 + (e x2 csc(3x) x 2 ) ( 2)[(1 + x 2 ) 3 ](1 + x 2 ) ((e x2 ) csc(3x) + e x2 (csc(3x)) 2x)(1 + x 2 ) 2 + (e x2 csc(3x) x 2 ) ( 2)[(1 + x 2 ) 3 ] (2x) ([e x2 (2x) csc(3x) + e x2 ( csc(3x) cot(3x)) 3] 2x)(1 + x 2 ) 2 + (e x2 csc(3x) x 2 ) ( 2)[(1 + x 2 ) 3 ](2x) (j) (x 4 e 3x cos(x)) (x 4 e 3x ) cos(x) + x 4 e 3x (cos(x)) [(x 4 ) e 3x + x 4 (e 3x ) ] cos(x) + x 4 e 3x ( sin(x)) [4x 3 e 3x + x 4 e 3x ( 3)] cos(x) + x 4 e 3x ( sin(x)) (k) Note tat sin (2x) x So ( sin (2x) x x cos3 (2x) 3 sin (2x)x x cos3 (2x) x cos3 (2x) 3 ) (sin (2x)x 1 ) 1 3 (x cos3 (2x)) (sin (2x)) x 1 + sin (2x)(x 1 ) 1 3 (x cos 3 (2x) + x(cos 3 (2x)) ) ( ) (sin 6 (2x))(sin(2x)) x 1 +sin (2x) ( 1)x (cos3 (2x)+x 3(cos 2 (2x))(cos(2x)) ) ( ) (sin 6 (2x))(cos(2x)) 2 x 1 + sin (2x) ( 1)x (cos3 (2x) + x 3(cos 2 (2x))( 2 sin(2x)) 2) (l) Note tat So ( 1 + t cos(t 2 ) 2t3 3 sin(t2 ) [1 + t cos(t 2 ) 2t3 3 sin(t2 )] t cos(t 2 ) 2t3 3 sin(t2 )) ([1 + t cos(t 2 ) 2t3 3 sin(t2 )] 1 2 ) 1[1 + t 2 cos(t2 ) 2t3 3 sin(t2 )] 1 2 [1 + t cos(t 2 ) 2t3 3 sin(t2 )] 1[1 + t 2 cos(t2 ) 2t3 3 sin(t2 )] 1 2 [t cos(t 2 )+t(cos(t 2 )) ( 2t3 3 ) sin(t 2 ) 2t3 3 (sin(t2 )) ]

4 Solution to Review Problems for Midterm II MATH 180: page 4 of 6 1[1 + t 2 cos(t2 ) 2t3 3 sin(t2 )] 1 2 [cos(t 2 ) + t( sin(t 2 )) (2t) (2t 2 ) sin(t 2 ) 2t 3 3 (cos(t2 )) 2t] 4 Find te first derivative (y ) and second derivative (y ) of te following functions (a) y (6 + 4 x ) (b) y x 3 e 3x Solution: (a)note tat y (6 + 4 x ) (6 + 4x 1 ) So y [(6 + 4x 1 ) ] (6 + 4x 1 ) 4 (6 + 4x 1 ) (6 + 4x 1 ) 4 ( 4x 2 ) 20(6 + 4x 1 ) 4 x 2 y [ 20(6 + 4x 1 ) 4 x 2 ] 20[(6 + 4x 1 ) 4 ] x 2 20(6 + 4x 1 ) 4 (x 2 ) 20 4 (6 + 4x 1 ) 3 ((6 + 4x 1 )) x 2 20(6 + 4x 1 ) 4 ( 2x 3 ) 20 4 (6 + 4x 1 ) 3 ( 4x 2 ) x (6 + 4x 1 ) 4 x (6 + 4x 1 ) 3 x (6 + 4x 1 ) 4 x 3 (b) y (x 3 e 3x ) (x 3 ) e 3x + x 3 (e 3x ) 3x 2 e 3x + x 3 e 3x 3 3x 2 e 3x + 3x 3 e 3x (3x 2 + 3x 3 )e 3x y [(3x 2 + 3x 3 )e 3x ] (3x 2 + 3x 3 ) e 3x + (3x 2 + 3x 3 )(e 3x ) (6x + 9x 2 )e 3x + (3x 2 + 3x 3 )e 3x 3 (6x + 9x 2 )e 3x + (9x 2 + 9x 3 )e 3x (6x + 9x 2 + 9x 2 + 9x 3 )e 3x (6x + 18x 2 + 9x 3 )e 3x Use implicit differentiation to find dy dx (a) 2xy y 2 x (b) x 3 + 3x 2 y + y 3 8 (c) x+y x y x2 + y 2 (d) cos(xy) + x y (e) e xy sin(x + y) In implicit differentiation, Suppose y y(x) ten (f(y)) f (y)y Solution:(a) Differentiating 2xy y 2 x, we get 2(xy) (y 2 ) x 2x y+2xy 2yy 1 2y+2xy 2yy 1 Now we ave 2xy 2yy 1 2y y (2x 2y) 1 2y y 1 2y 2x 2y (b) Differentiating te equation, we get (x 3 + 3x 2 y + y 3 ) (8) 3x 2 + 3(x 2 ) y + 3x 2 y + 3y 2 y 0 and 3x 2 + 6xy + 2x 2 y + 3y 2 y 0 Tis implies tat 2x 2 y + 3y 2 y 3x 2 6xy, y (2x 2 + 3y 2 ) 3x 2 6xy and y 3x2 6xy 2x 2 +3y 2 (c) Note tat x+y x y x2 + y 2 is te same as (x + y) (x y)(x 2 y 2 ) Differentiating te equation, we get (x + y) [(x y)(x 2 y 2 )] 1 + y (x y) (x 2 y 2 ) + (x y)(x 2 y 2 ) 1 + y (1 y )(x 2 y 2 ) + (x y)(2x 2yy ) 1 + y x 2 y 2 y (x 2 y 2 ) + 2x(x y) 2y(x y)y y + y (x 2 y 2 ) + 2y(x y)y x 2 y 2 + 2x 2 2xy 1 3x 2 y 2 2xy 1 y (1 + x 2 y 2 + 2yx 2y 2 ) 3x 2 y 2 2xy 1 and y (1 + x 2 3y 2 + 2yx) 3x 2 y 2 2xy 1 Tis gives y 3x2 y 2 2xy 1 1+x 2 3y 2 +2yx

5 MATH 180: page of 6 Solution to Review Problems for Midterm II (d) [cos(xy) + x ] (y ) sin(xy)(xy) + x 4 y 4 y sin(xy)(y + xy ) + x 4 y 4 y sin(xy)y sin(xy)xy + x 4 y 4 y sin(xy)xy y 4 y sin(xy)y x 4 y ( sin(xy)x y 4 ) sin(xy)y x 4 y sin(xy)y x4 sin(xy)x y 4 (e) (e xy ) (sin(x + y)) e xy (xy) cos(x + y)(x + y) e xy (y+xy ) cos(x+y)(1+y ) e xy y+e xy xy cos(x+y)+ cos(x+y)y cos(x + y)y + e xy xy cos(x + y) e xy y y ( cos(x + y) + e xy x) cos(x + y) e xy y y cos(x+y) exy y cos(x+y)+e xy x 6 Sow tat (1, 2) lie on te curve 2x 3 + 2y 3 9xy 0 Ten find te te tangent and normal to te curve at (1, 2) Solution: Plugging (1, 2) to te equation 2x 3 + 2y 3 9xy, we get Tis means tat (1, 2) lie on te curve 2x 3 + 2y 3 9xy 0 Next we find y by implicit differentiation Differentiating 2x 3 + 2y 3 9xy 0, we get 2(x 3 ) + 2(y 3 ) 9(xy) 0 6x 2 + 6y 2 y 9y 9xy 0 6y 2 y 9xy 6x 2 + 9y y (6y 2 9x) 6x 2 + 9y y 6x2 +9y 6y 2 9x At (1, 2), we ave y (1) So te slope of te tangent line is m 4 and te point is (1, 2) By te point slope formula, we ave y 2 4 (x 1) From te slope of te tangent line, we know tat te slope of te normal line is m 4 So te equation of te normal line is y 2 (x 1) 4 Find te normal to te curve xy + 2x y 0 tat are parallel to te line x + 2y 0 Solution: First we find y by implicit differentiation Differentiating xy + 2x y 0, we get (xy) + 2(x) (y) 0 y + xy + 2 y 0 xy y y 2 So te slope of te tangent line at (x, y) is y 2 From ere, we know tat te slope of te normal line is x 1 x 1 x 1 Te equation x + 2y 0 can be rewritten as 2y x and y 2 y+2 y 1x So te slope is m 1 At (x, y), te slope of te normal to 2 2 te curve xy + 2x y 0 tat are parallel to te line x + 2y 0 must ave slope 1 x 1 Tis implies tat 1, 2x 2 y 2 and y 2x 2 y+2 2 y (x 1) y 2 y y 2 x 1

6 Solution to Review Problems for Midterm II MATH 180: page 6 of 6 Plugging y 2x into te equation of te curve xy + 2x y 0, we get x( 2x) + 2x ( 2x) 0 2x 2 + 2x + 2x 0 2x 2 + 4x 0 2x(x 2) 0 x 0or x 2 Recall tat y 2x Tis implies tat y 0 or y 4 So te point is (0, 0) and te slope is (2, 4) Recall te slope of te normal line is 1 So te normal line parallel to x + 2y 0 2 are y 1x (point(0, 0))and y (x 2) (point(2, 4)) 2 2

Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x

Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc

More information

Math 250 Skills Assessment Test

Math 250 Skills Assessment Test Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).

More information

5, tan = 4. csc = Simplify: 3. Simplify: 4. Factor and simplify: cos x sin x cos x

5, tan = 4. csc = Simplify: 3. Simplify: 4. Factor and simplify: cos x sin x cos x Precalculus Final Review 1. Given the following values, evaluate (if possible) the other four trigonometric functions using the fundamental trigonometric identities or triangles csc = - 3 5, tan = 4 3.

More information

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x). You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and

More information

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever

More information

Math 229 Mock Final Exam Solution

Math 229 Mock Final Exam Solution Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it

More information

Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives

Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives Topic 4 Outline 1 Derivative Rules Calculating the Derivative Using Derivative Rules Implicit Functions Higher-Order Derivatives D. Kalajdzievska (University of Manitoba) Math 1500 Fall 2015 1 / 32 Topic

More information

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.

Mth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework. For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.

More information

UNIT-IV DIFFERENTIATION

UNIT-IV DIFFERENTIATION UNIT-IV DIFFERENTIATION BASIC CONCEPTS OF DIFFERTIATION Consider a function yf(x) of a variable x. Suppose x changes from an initial value x 0 to a final value x 1. Then the increment in x defined to be

More information

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Name Date Period. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. AB Fall Final Exam Review 200-20 Name Date Period MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) The position of a particle

More information

MATH 408N PRACTICE FINAL

MATH 408N PRACTICE FINAL 05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results

More information

Solving Equations. Pure Math 30: Explained! 255

Solving Equations. Pure Math 30: Explained!   255 Solving Equations Pure Math : Explained! www.puremath.com 55 Part One - Graphically Solving Equations Solving trigonometric equations graphically: When a question asks you to solve a system of trigonometric

More information

13 Implicit Differentiation

13 Implicit Differentiation - 13 Implicit Differentiation This sections highlights the difference between explicit and implicit expressions, and focuses on the differentiation of the latter, which can be a very useful tool in mathematics.

More information

MATH 101: PRACTICE MIDTERM 2

MATH 101: PRACTICE MIDTERM 2 MATH : PRACTICE MIDTERM INSTRUCTOR: PROF. DRAGOS GHIOCA March 7, Duration of examination: 7 minutes This examination includes pages and 6 questions. You are responsible for ensuring that your copy of the

More information

Math 5 Trigonometry Chapter 4 Test Fall 08 Name Show work for credit. Write all responses on separate paper. Do not use a calculator.

Math 5 Trigonometry Chapter 4 Test Fall 08 Name Show work for credit. Write all responses on separate paper. Do not use a calculator. Math 5 Trigonometry Chapter Test Fall 08 Name Show work for credit. Write all responses on separate paper. Do not use a calculator. 23 1. Consider an arclength of t = travelled counter-clockwise around

More information

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained. Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive

More information

INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1.

INVERSE FUNCTIONS DERIVATIVES. terms on one side and everything else on the other. (3) Factor out dy. for the following functions: 1. INVERSE FUNCTIONS DERIVATIVES Recall the steps for computing y implicitly: (1) Take of both sies, treating y like a function. (2) Expan, a, subtract to get the y terms on one sie an everything else on

More information

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12

Grade: The remainder of this page has been left blank for your workings. VERSION E. Midterm E: Page 1 of 12 First Name: Student-No: Last Name: Section: Grade: The remainder of this page has been left blank for your workings. Midterm E: Page of Indefinite Integrals. 9 marks Each part is worth 3 marks. Please

More information

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.

Solution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4. December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need

More information

In general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f.

In general, if we start with a function f and want to reverse the differentiation process, then we are finding an antiderivative of f. Math 1410 Worksheet #27: Section 4.9 Name: Our final application of derivatives is a prelude to what will come in later chapters. In many situations, it will be necessary to find a way to reverse the differentiation

More information

Review Problems for Test 1

Review Problems for Test 1 Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And

More information

MATH 1A Midterm Practice September 29, 2014

MATH 1A Midterm Practice September 29, 2014 MATH A Midterm Practice September 9, 04 Name: Problem. (True/False) If a function f : R R is injective, ten f as an inverse. Solution: True. If f is injective, ten it as an inverse since tere does not

More information

dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)

dx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy) Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =

More information

Calculus Midterm Exam October 31, 2018

Calculus Midterm Exam October 31, 2018 Calculus Midterm Exam October 31, 018 1. Use l Hôpital s Rule to evaluate the it, if it exists. tan3x (a) (6 points) sinx tan3x = 0, sinx = 0, and both tan3x and sinx are differentiable near x = 0, tan3x

More information

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions

Spring 2015, MA 252, Calculus II, Final Exam Preview Solutions Spring 5, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card,

More information

Trigonometric Functions () 1 / 28

Trigonometric Functions () 1 / 28 Trigonometric Functions () 1 / 28 Trigonometric Moel On a certain ay, ig tie at Pacific Beac was at minigt. Te water level at ig tie was 9.9 feet an later at te following low tie, te tie eigt was 0.1 ft.

More information

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ). Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate

More information

Department of Mathematics, K.T.H.M. College, Nashik F.Y.B.Sc. Calculus Practical (Academic Year )

Department of Mathematics, K.T.H.M. College, Nashik F.Y.B.Sc. Calculus Practical (Academic Year ) F.Y.B.Sc. Calculus Practical (Academic Year 06-7) Practical : Graps of Elementary Functions. a) Grap of y = f(x) mirror image of Grap of y = f(x) about X axis b) Grap of y = f( x) mirror image of Grap

More information

Math 251, Spring 2005: Exam #2 Preview Problems

Math 251, Spring 2005: Exam #2 Preview Problems Math 5, Spring 005: Exam # Preview Problems. Using the definition of derivative find the derivative of the following functions: a) fx) = e x e h. Use the following lim =, e x+h = e x e h.) h b) fx) = x

More information

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where

2. Which of the following is an equation of the line tangent to the graph of f(x) = x 4 + 2x 2 at the point where AP Review Chapter Name: Date: Per: 1. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the

More information

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?

(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a? Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is

More information

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx

Sec. 14.3: Partial Derivatives. All of the following are ways of representing the derivative. y dx Math 2204 Multivariable Calc Chapter 14: Partial Derivatives I. Review from math 1225 A. First Derivative Sec. 14.3: Partial Derivatives 1. Def n : The derivative of the function f with respect to the

More information

Algebra 2/Trig AIIT.17 Trig Identities Notes. Name: Date: Block:

Algebra 2/Trig AIIT.17 Trig Identities Notes. Name: Date: Block: Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Name: Date: Block: Trigonometric Identities When two trig expressions can be proven to be equal to each other, the statement is called a trig identity

More information

(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2

(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2 Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,

More information

4 Partial Differentiation

4 Partial Differentiation 4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which

More information

Pre-Calculus II: Trigonometry Exam 1 Preparation Solutions. Math&142 November 8, 2016

Pre-Calculus II: Trigonometry Exam 1 Preparation Solutions. Math&142 November 8, 2016 Pre-Calculus II: Trigonometry Exam 1 Preparation Solutions Math&1 November 8, 016 1. Convert the angle in degrees to radian. Express the answer as a multiple of π. a 87 π rad 180 = 87π 180 rad b 16 π rad

More information

Topics and Concepts. 1. Limits

Topics and Concepts. 1. Limits Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits

More information

UNIT 3: DERIVATIVES STUDY GUIDE

UNIT 3: DERIVATIVES STUDY GUIDE Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average

More information

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,

1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2, 1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)

More information

Antiderivatives. Mathematics 11: Lecture 30. Dan Sloughter. Furman University. November 7, 2007

Antiderivatives. Mathematics 11: Lecture 30. Dan Sloughter. Furman University. November 7, 2007 Antiderivatives Mathematics 11: Lecture 30 Dan Sloughter Furman University November 7, 2007 Dan Sloughter (Furman University) Antiderivatives November 7, 2007 1 / 9 Definition Recall: Suppose F and f are

More information

Math 147 Exam II Practice Problems

Math 147 Exam II Practice Problems Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab

More information

Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018

Math 1552: Integral Calculus Final Exam Study Guide, Spring 2018 Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement

More information

1.3 Basic Trigonometric Functions

1.3 Basic Trigonometric Functions www.ck1.org Chapter 1. Right Triangles and an Introduction to Trigonometry 1. Basic Trigonometric Functions Learning Objectives Find the values of the six trigonometric functions for angles in right triangles.

More information

University Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.

University Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c. MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,

More information

1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.

1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x. Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of

More information

Final exam for MATH 1272: Calculus II, Spring 2015

Final exam for MATH 1272: Calculus II, Spring 2015 Final exam for MATH 1272: Calculus II, Spring 2015 Name: ID #: Signature: Section Number: Teaching Assistant: General Instructions: Please don t turn over this page until you are directed to begin. There

More information

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that

More information

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ

True or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.

More information

f(x) f(a) Limit definition of the at a point in slope notation.

f(x) f(a) Limit definition of the at a point in slope notation. Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point

More information

TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER

TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER Prof. Israel N. Nwaguru MATH 11 CHAPTER,,, AND - REVIEW WORKOUT EACH PROBLEM NEATLY AND ORDERLY ON SEPARATE SHEET THEN CHOSE THE BEST ANSWER TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER

More information

TRIGONOMETRY OUTCOMES

TRIGONOMETRY OUTCOMES TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES The functions that we have met so far can be described by expressing one variable explicitly in terms of another variable. y For example,, or y = x sin x,

More information

Differentiation Rules and Formulas

Differentiation Rules and Formulas Differentiation Rules an Formulas Professor D. Olles December 1, 01 1 Te Definition of te Derivative Consier a function y = f(x) tat is continuous on te interval a, b]. Ten, te slope of te secant line

More information

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions

Fall 2016, MA 252, Calculus II, Final Exam Preview Solutions Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and

More information

Ma 221 Homework Solutions Due Date: January 24, 2012

Ma 221 Homework Solutions Due Date: January 24, 2012 Ma Homewk Solutions Due Date: January, 0. pg. 3 #, 3, 6,, 5, 7 9,, 3;.3 p.5-55 #, 3, 5, 7, 0, 7, 9, (Underlined problems are handed in) In problems, and 5, determine whether the given differential equation

More information

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29

Practice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29 Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)

More information

FAIRFIELD COUNTY MATH LEAGUE (FCML) Match 4 Round 1 Arithmetic: Basic Statistics

FAIRFIELD COUNTY MATH LEAGUE (FCML) Match 4 Round 1 Arithmetic: Basic Statistics Match 4 Round 1 Arithmetic: Basic Statistics 1.) 6.4.) 14706.) 85 1.)The geometric mean of the numbers x 1, x, x n is defined to be n x 1 x...x n. What is the positive difference between the arithmetic

More information

The goal of today is to determine what u-substitution to use for trigonometric integrals. The most common substitutions are the following:

The goal of today is to determine what u-substitution to use for trigonometric integrals. The most common substitutions are the following: Trigonometric Integrals The goal of today is to determine what u-substitution to use for trigonometric integrals. The most common substitutions are the following: Substitution u sinx u cosx u tanx u secx

More information

Chapter 1. Functions 1.3. Trigonometric Functions

Chapter 1. Functions 1.3. Trigonometric Functions 1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius

More information

Core Mathematics 3 Differentiation

Core Mathematics 3 Differentiation http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative

More information

Questions from Larson Chapter 4 Topics. 5. Evaluate

Questions from Larson Chapter 4 Topics. 5. Evaluate Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of

More information

and verify that it satisfies the differential equation:

and verify that it satisfies the differential equation: MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people

More information

Chapter 2: Differentiation

Chapter 2: Differentiation Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L

More information

Unit 6 Trigonometric Identities

Unit 6 Trigonometric Identities Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations Prove trigonometric identities, using: Reciprocal identities Quotient identities Pythagorean identities Sum

More information

f(g(x)) g (x) dx = f(u) du.

f(g(x)) g (x) dx = f(u) du. 1. Techniques of Integration Section 8-IT 1.1. Basic integration formulas. Integration is more difficult than derivation. The derivative of every rational function or trigonometric function is another

More information

Math 152 Take Home Test 1

Math 152 Take Home Test 1 Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I

More information

Math 121. Exam II. November 28 th, 2018

Math 121. Exam II. November 28 th, 2018 Math 121 Exam II November 28 th, 2018 Name: Section: The following rules apply: This is a closed-book exam. You may not use any books or notes on this exam. For free response questions, you must show all

More information

Math 1310 Final Exam

Math 1310 Final Exam Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space

More information

MATH 151, SPRING 2018

MATH 151, SPRING 2018 MATH 151, SPRING 2018 COMMON EXAM II - VERSIONBKEY LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF

More information

2.3 More Differentiation Patterns

2.3 More Differentiation Patterns 144 te derivative 2.3 More Differentiation Patterns Polynomials are very useful, but tey are not te only functions we need. Tis section uses te ideas of te two previous sections to develop tecniques for

More information

Final Exam. Math 3 December 7, 2010

Final Exam. Math 3 December 7, 2010 Final Exam Math 3 December 7, 200 Name: On this final examination for Math 3 in Fall 200, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle.

More information

Math 131 Exam 2 Spring 2016

Math 131 Exam 2 Spring 2016 Math 3 Exam Spring 06 Name: ID: 7 multiple choice questions worth 4.7 points each. hand graded questions worth 0 points each. 0. free points (so the total will be 00). Exam covers sections.7 through 3.0

More information

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 611b Assignment #6 Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 611b Assignment #6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find a formula for the function graphed. 1) 1) A) f(x) = 5 + x, x < -

More information

Math 180, Final Exam, Fall 2012 Problem 1 Solution

Math 180, Final Exam, Fall 2012 Problem 1 Solution Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.

More information

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier. The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then

More information

Inverse Trig Functions

Inverse Trig Functions 6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)

More information

You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need:

You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: You are being asked to create your own AP CALCULUS Survival kit. For the survival kit you will need: Index cards Ring (so that you can put all of your flash cards together) Hole punch (to punch holes in

More information

Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question

Multiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten

More information

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0

c) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0 Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to

More information

Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

More information

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =

Test 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 = Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve:

More information

Chapter 2: Differentiation

Chapter 2: Differentiation Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.)

Your signature: (1) (Pre-calculus Review Set Problems 80 and 124.) (1) (Pre-calculus Review Set Problems 80 an 14.) (a) Determine if each of the following statements is True or False. If it is true, explain why. If it is false, give a counterexample. (i) If a an b are

More information

Math 112 (Calculus I) Midterm Exam 3 KEY

Math 112 (Calculus I) Midterm Exam 3 KEY Math 11 (Calculus I) Midterm Exam KEY Multiple Choice. Fill in the answer to each problem on your computer scored answer sheet. Make sure your name, section and instructor are on that sheet. 1. Which of

More information

1 Solution to Homework 4

1 Solution to Homework 4 Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value

More information

This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.

This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set. Yanimov Almog WeBWorK assignment number Sections 3. 3.2 is ue : 08/3/207 at 03:2pm CDT. Te (* replace wit url for te course ome page *) for te course contains te syllabus, graing policy an oter information.

More information

Blue Pelican Calculus First Semester

Blue Pelican Calculus First Semester Blue Pelican Calculus First Semester Student Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function

More information

Math 222 Spring 2013 Exam 3 Review Problem Answers

Math 222 Spring 2013 Exam 3 Review Problem Answers . (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w

More information

1d C4 Integration cot4x 1 4 1e C4 Integration trig reverse chain 1

1d C4 Integration cot4x 1 4 1e C4 Integration trig reverse chain 1 A Assignment Nu Cover Sheet Name: Drill Current work Question Done BP Ready Topic Comment Aa C4 Integration Repeated linear factors 3 (x ) 3 (x ) + c Ab C4 Integration cos^ conversion x + sinx + c Ac C4

More information

Calculus I Review Solutions

Calculus I Review Solutions Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.

More information

MAS113 CALCULUS II SPRING 2008, QUIZ 5 SOLUTIONS. x 2 dx = 3y + y 3 = x 3 + c. It can be easily verified that the differential equation is exact, as

MAS113 CALCULUS II SPRING 2008, QUIZ 5 SOLUTIONS. x 2 dx = 3y + y 3 = x 3 + c. It can be easily verified that the differential equation is exact, as MAS113 CALCULUS II SPRING 008, QUIZ 5 SOLUTIONS Quiz 5a Solutions (1) Solve the differential equation y = x 1 + y. (1 + y )y = x = (1 + y ) = x = 3y + y 3 = x 3 + c. () Solve the differential equation

More information

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator. Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative

More information

Review for Exam IV MATH 1113 sections 51 & 52 Fall 2018

Review for Exam IV MATH 1113 sections 51 & 52 Fall 2018 Review for Exam IV MATH 111 sections 51 & 52 Fall 2018 Sections Covered: 6., 6., 6.5, 6.6, 7., 7.1, 7.2, 7., 7.5 Calculator Policy: Calculator use may be allowed on part of te exam. Wen instructions call

More information

MATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed

MATH 1241 FINAL EXAM FALL 2012 Part I, No Calculators Allowed MATH 11 FINAL EXAM FALL 01 Part I, No Calculators Allowed 1. Evaluate the limit: lim x x x + x 1. (a) 0 (b) 0.5 0.5 1 Does not exist. Which of the following is the derivative of g(x) = x cos(3x + 1)? (a)

More information

Chapter 2 Differentiation. 2.1 Tangent Lines and Their Slopes. Calculus: A Complete Course, 8e Chapter 2: Differentiation

Chapter 2 Differentiation. 2.1 Tangent Lines and Their Slopes. Calculus: A Complete Course, 8e Chapter 2: Differentiation Chapter 2 Differentiation 2.1 Tangent Lines and Their Slopes 1) Find the slope of the tangent line to the curve y = 4x x 2 at the point (-1, 0). A) -1 2 C) 6 D) 2 1 E) -2 2) Find the equation of the tangent

More information

Solutions to Exam 2, Math 10560

Solutions to Exam 2, Math 10560 Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If

More information

MA Practice Exam #2 Solutions

MA Practice Exam #2 Solutions MA 123 - Practice Exam #2 Solutions Name: Instructions: For some of the questions, you must show all your work as indicated. No calculators, books or notes of any form are allowed. Note that the questions

More information

June 9 Math 1113 sec 002 Summer 2014

June 9 Math 1113 sec 002 Summer 2014 June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the

More information