Midterm 1 Review Sheet PDF

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Review document and sheet containing practice questions...

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MATH 13300 Section 42 Midterm 1 Review Sheet Karl Schaefer April 14, 2018

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Computational Questions

Exercise 1.1. Evaluate each of the following expressions. Give answers in radians, not degrees. 1. arcsin 1 √ ! 2 2. arccos − 2 3. arctan

√

3

 π  4. arccos cos − 2 

Exercise 1.2. Evaluate each of the following derivatives. 1.

d arcsin(x2 ) dx

2.

d (arcsin x)2 dx

3.

d arctan(xex ) dx

4.

d sec−1 (ln x) dx

Exercise 1.3. Derive the following reduction formulas. Z Z − sinn−1 x cos x n − 1 n + 1. sin x dx = sinn−2 x dx n n Z Z cosn−1 x sin x n − 1 n + 2. cos x dx = cosn−2 x dx n n Exercise 1.4. Evaluate the following integrals. You may use the reduction formulas from the previous exercise if needed. 1

1.

Z

arctan x dx

2.

Z

sin4 x dx

3.

Z

1 √ dx x− x

4.

Z

√ x2 3 6x + 1 dx

5.

Z

√

6.

Z

x4 ln x dx

7.

Z

cos(x)ex dx

8.

Z

(ln x)2 dx

9.

Z

(ln x)3 dx

x2 dx 1 − x2

Exercise 1.5. Determine if each of the following sequences converge or diverge. If it converges, try to find its limit. (Not all of them will have limits you can actually compute!) 1. an =

1 n3/2

2. an = (−1)n 3. an =

n2 + (−1)n (π)−n 2n2 + 1

4. an =

n2 2n

5. an = n!/n 6. an =

32n (n!)2

Exercise 1.6. Determine if the each of the following series converge or diverge. If it converges, try to find its sum. 1.

∞ X n=1

n2 2n2 + 1

2

2.

∞  −n X 1 2 n=1

∞ X 3n+1 3. 5n n=1

4. 4 − 1 +

1 1 1 + ··· + − 4 16 64

∞ X 1 5. n n=1

6.

∞ X en−1 π n−1 n=1

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Conceptual Topics

Exercise 2.1. You should know and be familiar with the following: 1. The relationship between trig functions and right triangles. 2. How to find the derivatives of the inverse trig functions. 3. The definition of what it means for a sequence and a series to converge (and the differences between them!). 4. Properties of limits of sequences and sums of series (e.g. the limit of the sum is the sum of the limits). 5. The Monotonic Sequence Theorem. 6. Geometric sequences/series and when they converge/diverge. 7. The nth-term limit test for divergence of a series. Exercise 2.2. Show that each of the following equations is an identity. √ 1. sin(cos−1 x) = 1 − x2 √ 2. sec(tan−1 x) = 1 + x2 √ x2 − 1 3. cos(csc−1 x) = x Exercise 2.3. Let cot−1 (x) be the inverse function of cot(x). Use the method from class to find the derivative of cot−1 (x). Exercise 2.4. Review the “extra problems” that I put on each of the homework assignments!

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