MA 114 Worksheet # 1: Improper Integrals

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1 MA 4 Worksheet # : Improper Integrals. For each of the following, determine if the integral is proper or improper. If it is improper, explain why. Do not evaluate any of the integrals. (c) x x 2 5x + 6 dx 2x dx ln (x ) dx (d) (e) π/2 0 sin x + x 2 dx sec x dx 2. For the integrals below, determine if the integral is convergent or divergent. Evaluate the convergent integrals. 0 2x dx xe x2 dx (c) (d) 2 x 3 0 2x 3 dx (write the numerator as 2 (2x 3) 3 2 ) 0 sin θ dθ 3. Consider the improper integral x p dx. Integrate using the generic parameter p to prove the integral converges for p > and diverges for p. You will have to distinguish between the cases when p = and p when you integrate. 4. Use the Comparison Theorem to determine whether the following integrals are convergent or divergent. 2 + e x dx x x + x6 + x dx 5. Explain why the following computation is wrong and determine the correct answer. (Try sketching or graphing the integrand to see where the problem lies.) 0 2 2x 8 dx = u du = 2 2 ln x 4 = (ln 2 ln 4) 2 where we used the substitution u(x) = 2x 8 u(2) = 4, u(0) = 2 du dx = 2

2 MA 4 Worksheet # 2: Sequences. Write the first four terms of the sequences with the following general terms: n! 2 n n n + (c) ( ) n+ 2. Find a formula for the nth term of the sequence 3. Conceptual Understanding: What is a sequence? What does it mean to say that a sequence is bounded? {, 8, 27, } 64,.... (c) What does it mean to say that a sequence is defined recursively? (d) What does it mean to say that a sequence converges? 4. Let a 0 = 0 and a =. Write out the first five terms of {a n } where a n is recursively defined as a n+ = 3a n + a n Suppose that a sequence {a n } is bounded above and below. Does it converge? If not, produce a counterexample. 6. Show that the sequence with general term a n = 3n2 n {a n } converge? is increasing. Find and upper bound. Does 7. Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. a n =.0 n. b n = 3n2 +n+ 2n 2 3. (c) c n = e n2.

3 MA 4 Worksheet # 3: Summing an Infinite Series. (Review) Compute the following sums 5 3n 6 k=3 ( sin ( π ) 2 + πk) + 2k 2. Conceptual Understanding: What is a series? What is the difference between a sequence and a series? (c) What does it mean that a series converges? 3. Write the following in summation notation: Calculate S 3, S 4, and S 5 and then find the sum of the telescoping series S = 5. Use Theorem 3 of 0.2 (Divergence Test) to prove that the following two series diverge: n 0n + 2 n n2 + ( n + ). n Use the formula for the sum of a geometric series to find the sum or state that the series diverges and why: ( π ) n. e (c) (d) n 5 n.

4 MA 4 Worksheet # 4: Series with Positive Terms. Use the Integral Test to determine if the following series converge or diverge: (c) + n 2 n 2 e n3 n=2 n (n 2 + 2) 3/2 2. Show that the infinite series converges if p > and diverges otherwise by Integral Test. np 3. Use the Comparison Test (or Limit Comparison Test) to determine whether the infinite series is convergent or divergent. (c) (d) (e) (f) n 3/2 + 2 n n n 4 n n + n! n 4 n 2 (n + )!

5 MA 4 Worksheet # 5: Absolute and Conditional Convergence. Conceptual understanding: Let a n = n 3n +. Does {a n} converge? Does Give an example of a divergent series (c) Does there exist a convergent series a n converge? a n where lim n a n = 0. a n which satisfies lim n a n 0? Explain. (d) When does a series converge absolutely? When does a series converge conditionally? (e) State the Leibniz test for alternating series. 2. Determine whether the series is absolutely convergent, conditionally convergent, or divergent. (c) ( ) n+ 5 n ( ) n ln(n + ) 3 cos(5) n 3. Identify the following statements as true or false. If the statement is true, cite evidence from the text to support it. If the statement is false, correct it so that it is a true statement from the text. To prove that the series then the series converges. a n converges you should compute the limit lim n a n. If this limit is 0 One way to prove that a series is convergent is to prove that it is absolutely convergent. (c) An infinite series converges when the limit of the sequence of partial sums converges.

6 . State the Root Test. MA 4 Worksheet # 6: Ratio and Root Test State the Ratio Test. 2. Determine whether the series is convergent or divergent. ( 3n 3 ) n + 2n 4n 3 + (c) 2 n n 2 n! e n n! 3. Identify the following statements as true or false. If the statement is true, cite evidence from the text to support it. If the statement is false, correct it so that it is a true statement from the text. To apply the Ratio Test to the series a n you should compute lim n a n+. If this limit is less a n than then the series converges absolutely. n To apply the Root Test to the series a n you should compute lim an. If this limit is or n larger then the series diverges.

7 MA 4 Worksheet # 7: Power Series. Find the radius and interval of convergence for 2. Find the radius and interval of convergence for 4 3. Find the radius and interval of convergence for 4. Find the radius and interval of convergence for ( ) n n 4 n (x 3) n. 2 n n (4x 8)n. x 2n ( 3) n. n!(x 2) n. 5. Give the definition of the radius of convergence of a power series 6. Use term by term integration and the fact that a n x n dx = arctan(x) to derive a power series + x2 centered at x = 0 for the arctangent function. [Hint: + x 2 = ( x 2 ).] 7. Use the same idea as above to give a series expression for ln( + x), given that You will again want to manipulate the fraction 8. Write ( + x 2 ) 2 as a power series. [Hint: Use term by term differentiation.] + x = ( x) as above. dx = ln( + x). + x

8 . Evaluate 2. Evaluate 3. Evaluate MA 4 Worksheet # 8: Review for Exam e x dx dx x ln(x). x x 3 dx. 4. Determine the limit of the sequence or state that the sequence diverges. a n = n + 3 n b n = cos(n) n (c) c n = en + ( 3) n 5 n (d) d n = n /n [HINT: Verify that x /x = e ln(x) x and use l Hospital s Rule.] ( ) 5. Determine whether or not n 2 converges. 6. Evaluate n=3 [HINT: Use n(n + 3). n(n + 3) = 3 n=2 ( n ).] n Find a value of N such that S N approximates the series with an error of at most 0 5 where ( ) n+ S = n(n + 2)(n + 3) 8. Determine convergence or divergence. 5 n. (c) (d) (e) ( ) n n2 +. ne n2. e n n!. e n n n. 9. Determine the radius of convergence of the following series. 2 n x n n!

9 n=2 (2x 3) n n ln(n)

10 MA 4 Worksheet # 9: Taylor Series. Find the terms through degree 3 of the Maclaurin series of f(x). f(x) = ( + x) /4. f(x) = e sin(x). 2. Find the Taylor series centered at c and find the interval on which the expansion converges to f. f(x) = x at c =. f(x) = e 3x at c =. (c) f(x) = x 3 + 3x at c = 0. (d) f(x) = x 3 + 3x at c = Find a power series representation for f(x) = x cos(x 2 ). g(x) = ( + x)e x. e x cos(x) 4. Show that lim = using power series. Verify your answer with l Hospital s Rule. x 0 sin(x) [Hint: Write out the power series for each term and factor out the lowest power of x from the numerator and the denominator, and then consider the limit.]

11 MA 4 Worksheet # 0: Taylor polynomials and volumes of solids with known cross sections Recommendation: Drawing a picture of the solids may be helpful during this worksheet.. What is T 3 (x) centered at a = 3 for a function f(x) where f(3) = 9, f (3) = 8, f (3) = 4, and f (3) = 2? 2. Calculate the Taylor polynomials T 2 (x) and T 3 (x) centered at x = a for the given function and value of a. f(x) = tan x, a = π 4 f(x) = x 2 e x, a = (c) f(x) = ln x x, a = 3. Let T 2 (x) be the Taylor polynomial of f(x) = x at a = 4. Apply the error bound to find the maximum possible value of f(.) T 2 (.). Show that we can take K = e.. 4. Let f(x) = 3x 3 + 2x 2 x 4. Calculate T k (x) for k =, 2, 3, 4, 5 at both a = 0 and a =. Show that T 3 (x) = f(x) in both cases. Let T n (x) be the n th Taylor polynomial at x = a for a polynomial f(x) of degree n. Based on part, guess the value of f(x) T n (x). Prove that your guess is correct using the error bound. 5. Conceptual Understanding: If a solid has a cross-sectional area given by the function A(x), what integral should be evaluated to find the volume of the solid? 6. Let V be the volume of a right circular cone of height 0 whose base is a circle of radius 4. Use similar triangles to find the area of a horizontal cross section at a height y. Using this area, calculate V by integrating the cross-sectional area. 7. Calculate the volume of the following solid. The base is a square, one of whose sides is the interval [0, l] along the x-axis. The cross sections perpendicular to the x-axis are rectangles of height f(x) = x Calculate the volume of the following solid. The base is the region enclosed by y = x 2 and y = 3. The cross sections perpendicular to the y-axis are squares. 9. The base of a certain solid is the triangle with vertices at ( 0, 5), (5, 5), and the origin. Cross-sections perpendicular to the y-axis are squares. Find the volume of the solid. 0. As viewed from above, a swimming pool has the shape of the ellipse y2 =. The cross 600 sections perpendicular to the ground and parallel to the y-axis are squares. Find the total volume of the pool.. Calculate the volume of the following solid. The base is a circle of radius r centered at the origin. The cross sections perpendicular to the x-axis are squares. 2. Calculate the volume of the following solid. The base is the parabolic region {(x, y) x 2 y 4}. The cross sections perpendicular to the y-axis are right isosceles triangles whose hypotenuse lies in the region. x 2

12 MA 4 Worksheet # : Density and Average Value. Conceptual Understanding: If the linear mass density of a rod at position x is given by the function ρ(x), what integral should be evaluated to find the mass of the rod between points a and b? If the radial mass density of a disk centered at the origin is given by the function ρ(r), where r is the distance from the center point, what integral should be evaluated to find the mass of a disk of radius R? (c) Write down the equation for the average value of an integrable function f(x) on [a, b]. 2. Find the total mass of a -meter rod whose linear density function is ρ(x) = 0(x + ) 2 kg/m for 0 x Find the average value of the following functions over the given interval. f(x) = x 3, [0, 4] f(x) = x 3, [, ] [ (c) f(x) = cos(x), 0, π ] 6 (d) f(x) = x 2, [, ] + (e) f(x) = sin(π/x) x 2, [, 2] (f) f(x) = e nx, [, ] (g) f(x) = 2x 3 6x 2, [, 3] (h) f(x) = x n for n 0, [0, ] 4. Odzala National Park in the Republic of the Congo has a high density of gorillas. Suppose that the radial population density is ρ(r) = 52( + r 2 ) 2 gorillas per square kilometer, where r is the distance from a grassy clearing with a source of water. Calculate the number of gorillas within a 5 km radius of the clearing. 5. Find the total mass of a circular plate of radius 20 cm whose mass density is the radial function ρ(r) = cos(πr 2 ) g/cm 2.

13 Worksheet #2: Volumes of Revolution. Find the volume of the solid obtained by rotating the region bounded by y =, y = 0, x =, and x5 x = 6, about the x-axis. 2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = 0, y = cos(2x), x = π 2, x = 0 about the line y = Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curves x = 0, y =, x = y, about the line y =. 4. For each of the following, use disks or washers to find the integral expression for the volume of the region. R is the region bounded by y = x 2 and y = 0; about the x-axis. R is the region bounded by x = 2 y, x = 0, and y = 9; about the y-axis. (c) R is the region bounded by y = x 2 and y = 0; about the line y =. (d) Between the regions in part and part (c), which volume is bigger? Why? First argue without computing the integrals, then also evaluate the integrals to check your answer. (e) R is the region bounded by y = e x, y = and x = 2; about the line y = 2. (f) R is the region bounded by y = x and y = x; about the line x = Find the volume of the cone obtained by rotating the region in the first quadrant under the segment joining (0, h) and (r, 0) about the y-axis. 6. A soda glass has the shape of the surface generated by revolving the graph of y = 6x 2 for 0 x about the y-axis. Soda is extracted from the glass through a straw at the rate of /2 cubic inch per second. How fast is the soda level in the glass dropping when the level is 2 inches? (Answer should be implicitly in units of inches per second.) 7. The torus is the solid obtained by rotating the circle (x a) 2 + y 2 = b 2 around the y-axis (assume that a > b). Show that it has volume 2π 2 ab 2. [Hint: Draw a picture, set up the problem and evaluate the integral by interpreting it as the area of a circle.] 8. Conceptual understanding of disk and shell method: Write a general integral to compute the volume of a solid obtained by rotating the region under y = f(x) over the interval [a, b] about the y-axis using the method of cylindrical shells. If you use the disk method to compute the same volume, are you integrating with respect to x or y? Why? 9. Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the y-axis. y = 3x 2, y = 6 x, x = 0 y = x 2, y = 8 x 2, x = 0, for x 0

14 MA 4 Worksheet #3: Shell Method and Work. Sketch the enclosed region and use the Shell Method to calculate the volume of the solid when rotated about the x-axis. x = 4 y +, x = 3 4 y, y = 0 x = y(4 y), x = 0 2. Use both the Shell and Disk Methods to calculate the volume obtained by rotating the region under the graph of f(x) = 8 x 3 for 0 x 2 about: the x-axis the y-axis 3. Use the Shell method to find the volume obtained by rotating the region bounded by y = x 2 +2, y = 6, x = 0, and x = 2 about the following axes: x = 2 x = 3 4. Find the integral for the volume of the solid obtained by rotating f(x) = e x about the y-axis from 0 x 2. Do not evaluate the integral. This will be done on the next worksheet. 5. Conceptual Understanding: Define and describe work. What are its units? What is the difference between work and force? Determine the work done in lifting a kg weight through a distance of m near the surface of the earth, maintaining a constant velocity. (c) How much work is done in lifting a kg weight up m at a constant velocity and then lowering it back m at a constant velocity? 6. Determine the work done in lifting a 500 kg elevator 000 m to the top floor of a building. How much work is done lowering a 500 kg elevator 000 m from the top floor of a building to the ground floor? How much work is done making the round trip? 7. A force of 50 N will stretch a spring from its natural length of 5 cm to 5 cm. How much work will be done in stretching the spring from 5 cm to 30 cm? 8. Calculate the work against gravity required to build a right circular cone of height 4 m and base radius.2 m out of a lightweight material of density 600 kg/m Consider a rectangular tank of water that is 5 meters tall and has a base of size 8 4 meters. It has a spout on its top surface. Calculate the work required to pump all of the water out of the tank. Dimensions are in meters, and the density of water is 000 kg/m 3.

15 Math 4 Worksheet # 4: Integration by Parts and Trigonometric Integrals. Use the product rule to find (u(x)v(x)). Next use this result to prove integration by parts, namely that u(x)v (x)dx = u(x)v(x) v(x)u (x) dx. 2. Which of the following integrals should be solved using substitution and which should be solved using integration by parts? x cos(x 2 ln (arctan(x)) ) dx, (c) + x 2 dx, e x sin(x) dx, (d) xe x2 dx Using these examples, try and formulate a general rule for when integration by parts should be used as opposed to substitution. 3. Solve the following integrals using integration by parts: x 2 sin(x) dx, (d) (2x + )e x dx, (e) (c) x sin (3 x) dx, 2x arctan(x) dx, ln(x) dx 4. Prove the reduction formula x n e x dx = x n e x n x n e x dx. Use this to evaluate x 3 e x dx. 5. Let f(x) be a twice differentiable function with f(0) = 6, f() = 5, and f () = 2. Evaluate 0 xf (x) dx. 6. Evaluate the following integrals. cos 2 (x) dx. (d) x 2 cos(x) dx. π/2 sin 2 (x) cos 2 (x) dx. (e) e x cos(x) dx. 0 (c) sin 3 (x) cos 2 (x) dx. 7. Evaluate sin(x) cos(x) dx by four methods the substitution u = cos(x), the substitution u = sin(x), (c) the identity sin(2x) = 2 sin(x) cos(x), (d) integration by parts. 8. Find the volume of the solid obtained by rotating f(x) = e x about the y-axis from 0 x 2.

16 Math 4 Worksheet # 5: Trigonometric Integrals. Evaluate the following integrals. tan 2 (x) dx sin(ϕ) cos 3 (ϕ) dϕ (c) tan 5 (x) sec 3 (x) dx (d) sin(8x) cos(5x) dx (e) (f) tan 2 (x) sec 2 dx (x) x sec 2 (x 2 ) tan 4 (x 2 ) dx (g) (h) π/4 π/4 π/2 π/4 tan 3 (x) dx cot 3 (x) dx 2. Prove that for any two integers m and n π 2π 0 cos(mx) cos(nx) dx = { 0 m n m = n.

17 . Expand 2 4 3x MA 4 Worksheet # 6: Review for Exam 2 into a power series with c = 0 and determine the radius of convergence. 2. Find the Taylor series centered at zero of the function f(x) = ln(x + 5). Find the Taylor series centered at zero of the function g(x) = x 3 ln(x 2 + 5). 3. Compute T 3 (x), the Taylor polynomial of the third order centered at x = 0, for f(x) = cos(x/π). 4. Compute T n (x), the Taylor polynomial of the nth order centered at x = 0, for f(x) = e 3x. 5. Let f(x) = e x. First compute T 3 (x) and then use the error bound to show that f(x) T 3 (x) x 4 /24 for all x Density and average value: Find the total mass of a circular plate of radius 20 cm whose mass density is the radial function ρ(r) = cos(πr 2 ) g/cm 2. Find the average value of f(x) = sin(x) cos(x) over [0, π]. 7. Volumes: (Disks) Let V be the volume of a right circular cone of height 0 whose base is a circle of radius 4. Use similar triangles to find the area of a horizontal cross section at a height y. Using this area, calculate the volume V by integrating the cross-sectional area. (Washers) Let R be a region bounded by y = x 2 and y =, if R is rotated about x-axis, what is the volume of the resulting solid? (c) (Cylindrical Shells) V is obtained by rotating the region under the graph y = 3x 2 for 0 x 2 about the y-axis. Calculate the volume of V. 8. Work: Calculate the work against gravity required to build a right circular cone of height 4 m and radius 2 m out of a lightweight material of density 600 kg/m 3. (See also question 7.) 9. Integration by Parts: x 2 cos(x) dx 2x arctan(x) dx 0. Trigonometric Integrals: sin 2 (x) cos 3 (x) dx tan 3 (x) sec 3 (x) dx

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