MAT136H1F - Calculus I (B) Long Quiz 1. T0101 (M3) Time: 20 minutes. The quiz consists of four questions. Each question is worth 2 points. Good Luck!

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1 MAT36HF - Calculus I (B) Log Quiz. T (M3) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each questio is worth 2 poits. Good Luck!. Write the itegral cos() d as a limit of Riema sums. Solutio: Notice that a = ad b =. We will partitio the iterval i pieces of equal legth. Therefore, = b a ad i = a + i =. I cosequece, ( ) i cos() d = lim cos( i ) = lim cos 2. What is the average value of f() = cos() o the iterval [ π, π]. Solutio: Notice that f( ) = cos( ) = cos() = f(). Therefore, f is a odd fuctio. Hece, π cos() d =. π Thus, f avg = π cos() d =. 2π π

2 ta 3. Fid 2 + d. Solutio: Let u = ta (), the du = ta 2 + d = d d. Therefore, + 2 u du = u2 u + C = ( ta () ) 2 + C Fid the area of the regio bouded by the curves y = 3 ad y = 3 2. Solutio: We first eed to fid the itersectio poits. For that, we solve the equatio 3 = = ( ) = = or = Graph: y y = 3 2 y = 3 So the area is A = (3 3 2 ) d = [ ] = 3 2 = 2.

3 MAT36HF - Calculus I (B) Log Quiz. T2 (R4) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each questio is worth 2 poits. Good Luck!. Write the limit lim 2 + i as a itegral. Solutio: The epressio suggests that = b a. Sice =, we ca try a = ad b =. The i = a + i = i. Therefore, our fuctio must be Hece, lim i= f() = i = 2 + d. 2. Compute cos 3 si d. Solutio: Let u = cos(), the du = si() d. Therefore, cos 3 () si() d = u 3 du = u4 4 + C = () cos4 + C. 4

4 3. Compute e d. Solutio: We use itegratio by parts. Let u = ad dv = e d. So, du = d ad v = e So we have, e d = e e d = e e + C. 4. Fid the area of the regio bouded by the curves y = 2 ad y = 2. Solutio: We first fid the itersectio poits. For that, we solve the equatio Graph: 2 = = ( + 2) = = or = 2. y y = 2 y = 2 2 Hece, the area is A = 2 ( 2 2 ) d = ] [ 2 3 = = 4 3.

5 MAT36HF - Calculus I (B) Log Quiz. T3 (T4) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each questio is worth 2 poits. Good Luck!. Write the itegral 2 d as a limit of Riema sums. Solutio: Notice that a = ad b = 2. We will partitio the iterval i pieces of equal legth. Therefore, = b a = 2 ad i = a + i = 2i. I cosequece, 2 d = lim ( i ) = lim ( 2i 2 ) Compute l d. Solutio: We use the substitutio u = l(). The du = d. Therefore, l d = du = l( u ) + C = l l() + C. u

6 3. Suppose that f is cotiuous ad 3 f() d = 2. What value must be take o by f o the iterval [, 3]? Eplai briefly. (Hit: Cosider the Mea Value Theorem for Itegrals) Solutio: By the Mea Value Theorem for itegrals, there must eist c [, 3] such that f(c) = f ave = 3 3 f() d = 2 2 =. Therefore, f must take o the value f(c) = for some c [, 3]. 4. Fid the area of the regio betwee the curves y = 4 ad y =. Solutio: We first fid the poits of itersectio. For that we eed to solve the equatio Graph: 4 = 4 = ( 3 ) = = or =. y y = 4 y = Hece, the area is A = [ ( 4 2 ) d = ] = 2 5 = 3.

7 MAT36HF - Calculus I (B) Log Quiz. T5 (T5) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each questio is worth 2 poits. Good Luck!. Evaluate the limit lim 2 + 2i. Solutio: The epressio suggests that = 2 b a. Sice = = 2, we ca try a = ad b = 2. So i = a + i = + i 2. Therefore, our fuctio must be Hece, = 2i lim f() = i = + d. 2. What is the average value of f() = si() cos() o the iterval [ π, π]. Solutio: Notice that f( ) = si( ) cos( ) = si() cos() = f(). Therefore, f is a odd fuctio. Hece, f ave = π si() cos() d =. 2π π

8 3. Fid sec 3 ta d. Solutio: Let u = sec(). The du = sec() ta()d. Therefore, sec 3 () ta() d = sec 2 () sec() ta() d = u 2 du = u3 + C 3 = sec3 () + C Fid the area of the regio bouded by the curves y = 2 ad y = + 2. Solutio: We first fid the poits of itersectio. For that, we eed to solve the equatio Graph: 2 = = ( 2)( + ) = = 2 or =. y y = 2 y = Hece, the area is A = 2 [ ( ) d = 2 ] = = 9 2.

9 MAT36HF - Calculus I (B) Log Quiz. T52 (R5) Time: 2 miutes Last Name: Studet ID: First Name: Please mark your tutorial sectio: T (M3) T2 (R4) T3 (T4) T5 (T5) T52 (R5) The quiz cosists of four questios. Each questio is worth 2 poits. Good Luck!. Evaluate the limit lim ( ) iπ π si. Solutio: The epressio suggests that = π b a. Sice = = π, we ca try a = ad b = π. So i = a + i = + i π = i π. Therefore, our fuctio must be Hece, lim si f() = si(). ( ) iπ π π = si() d = [ cos()] π = Cosider the fuctio f() = 3 2. Fid the umber c [, 2] that satisfies the Mea Value Theorem for itegrals o [, 2]. Solutio: By the Mea value Theorem for itegrals we kow that there eists c [, 2] such that f(c) = f ave = d = [ ] = 4. Hece, f(c) = 4 3c 2 = 4 c 2 = 4 3 c = 2 3 or c = 2 3. But oly c = 2 3 [, 2].

10 e 3. Fid d. Solutio: Let u =. The du = 2 d. Therefore, e d = 2 e u du = 2e u + C = 2e + C. 4. Fid the area of the regio bouded by the curves y = 2 ad y = 2 3. Solutio: We first fid the poits of itersectio. For that we eed to solve the equatio Graph: 2 = = ( + 3)( ) = = or = y y = 2 3 y = 2 Hece, the area is A = 3 ( ) d = ] [ = (9 9 9) = 32 3.

MATH Exam 1 Solutions February 24, 2016

MATH Exam 1 Solutions February 24, 2016 MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [