Calculus

Objectives: Students will:

1. evaluate derivatives for complexly constructed elementary functions;
2. evaluate definite and indefinite integrals; and
3. evaluate limits using algebraic, geometric, analytic techniques.

2. Learning Goal: Mathematics majors will learn and retain basic knowledge in the core branches of mathematics.

Objectives: Students will, during their senior year:

1. demonstrate proficiency in calculus;
2. demonstrate proficiency in linear algebra; and
3. demonstrate proficiency in algebra.

3. Learning Goal: Mathematics majors will be able to learn and explain mathematics on their own.

Objectives: Students will:

1. read a mathematics journal article and explain it, orally or in writing, to an audience of math majors and
2. after graduation, be able to master new mathematics necessary for their employment.

4. Learning Goal: Mathematics majors will be able to read and construct rigorous proofs.

Objectives: Students will:

1. construct clearly written proofs which use correct terminology and cite previous theorems;
2. construct proofs using mathematical induction;
3. construct proofs by contradiction; and
4. judge whether a proof is sound, and identify errors in a faulty proof.

5. Learning Goal: Mathematics majors will be able to obtain employment in their area of mathematical interest or gain admittance to a graduate program in mathematics.

Objectives: Students who:

1. seek admission to graduate schools in mathematics will succeed in gaining admission, and perform adequately in these programs;
2. seek entry-level employment in math-related fields will obtain it;
3. specialize in actuarial science will obtain entry-level work as actuaries, if they seek it;
4. specialize in secondary education will demonstrate proficiency in mathematics needed to obtain Initial Certification in New York State; or
5. seek jobs in secondary or elementary education will obtain jobs at the appropriate grade level.

6. Learning Goal: Master's students will recognize connections between different branches of mathematics.

Learning Objectives: Students will:

1. correctly incorporate specific examples from one branch of mathematics into their study of another branch of mathematics (e.g., Lp-spaces as an example in linear algebra) and
2. identify and explain cases in which major results of one branch of mathematics rely nontrivially on results from another branch (e.g., the application of linear algebra to solving systems of differential equations).

7. Learning Goal: Graduating master's degree students will be able to obtain employment in their area of mathematical interest or gain admittance to a doctoral program in mathematics.

Learning Objectives: Students who:

1. seek admission to doctoral programs in mathematics, applied mathematics, mathematical finance, mathematics education or other math-related fields will succeed in gaining admission to such programs, and perform adequately in these programs;
2. seek employment as full-time instructors at community colleges or as part-time instructors at four-year colleges or universities will obtain it; and
3. seek employment in other math-related fields will obtain it.

The equation is exactly what you just said: " 12 + y = 22 " .

As long as we're both here, let's go ahead and find the solution to it.

It could get complicated. Hold on to your hat, and try to stay with me:

12 + y = 22

Subtract 12 from each side:

y = 10

If you look in most calculus books there is a table of intergals. Your problem is one of the trigonometric forms. Below is given the reduction formula,

∫sinn(u) du = (-1/n)sinn-1(u)cos(u) + ((n-1)/n)∫sinn-2(u) du

where n ≥ 2 is an integer.

Basically yours would be n=7 & u=3x. You keep integrating until you lose the last integral sign on the right side. Pretty tedious work, but if I am using the right integral form & didn't make any mistakes the answer should be:

(-1/7)sin6(3x)cos(3x) - (6/35)sin4(3x)cos(3x) - (8/35)sin2(3x)cos(3x) - (16/35)cos(3x) + C

Here is the work:

Let u = 3x

u' = 3 dx, so that dx = du/3

∫sin7 3x dx = (1/3)∫sin7(u) du

∫sin7(u) du = (-1/7)sin6(u)cos(u) + (6/7)∫sin5(u) du

= (-1/7)sin6(u)cos(u) + (6/7)[(-1/5)sin4(u)cos(u) + (4/5)∫sin3(u) du]

= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) + (24/35)∫sin3(u) du

= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) + (24/35)[(-1/3)sin2(u)cos(u) + (2/3)∫sin(u) du]

= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) + (16/35)∫sin(u) du

= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) - (16/35)cos(u) + C

so that

∫sin7 3x dx = (1/3)∫sin7(u) du

= (1/3 [(-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) - (16/35)cos(u)] + C

= -(1/21)sin6(u)cos(u) - (2/35)sin4(u)cos(u) - (8/105)sin2(u)cos(u) - (16/105)cos(u) + C

= -(1/21)sin6(3x)cos(3x) - (2/35)sin4(3x)cos(3x) - (8/105)sin2(3x)cos(3x) - (16/105)cos(3x) + C

Or

∫sin7 3x dx = ∫sin63x sin 3x dx = ∫(sin2 3x)3 sin 3x dx =∫(1 - cos2 3x)3 sin 3x dx

Let u = cos 3x

u' = (cos 3x)'

du = -sin 3x*3 dx, and dx = du/-3sin 3x

= (-1/3)∫(1 - u2)3 du = ∫(1 - 3u2 + 3u4 + u6) du = (-1/3) [u - (3/3)u3 + (3/5)u5 - u7/7] + C

= (-1/3)u + (1/3)u3 - (1/5)u5 + (1/21)u7 + C

= (-1/3)cos 3x + (1/3)cos3 3x - (1/5)cos5 3x + (1/21)cos7 3x + C

2x + 21 = 9 x - 7

Add 7 to each side

2x + 21 = 9x

Subtract 2x from each side

21 = 7x

Divide each side by 7

3 = x

Sorted.