Find the volume generated by rotating the region bounded by

Question

Question 1. Convert from degrees to radians.

330°

Question 2. Convert from radians to degrees.

13Π/18

Question 3. Find the exact value.

tan (Π/3)

Question 4. Find the exact value.

sin (4Π/3)

Question 5. If sin(x) = 1/3 and sec(y) = 5/3, where x and y lie between 0 and Π/2, evaluate sin(x + y).

Question 6. Find all values of x such that sin(2x) = sin(x) and 0 ≤ x ≤ 2Π. (Enter your answers as a comma-separated list.)
x =

Question 7. Sketch the graph of the function y = 2 + sin(2x) without using a calculator.

Question 8. Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.)

lim_{t → -2} (t⁴ -6)/(2t² - 3t + 4)

Question 9. A tank contains 9000 L of pure water. Brine that contains 35 g of salt per liter of water is pumped into the tank at a rate of 25 L/min. The concentration of salt after t minutes (in grams per liter) is

C(t) = 35t/(360 + t)

As t → ∞, what does the concentration approach?

Question 10. Find the horizontal and vertical asymptotes of the curve.

y = x² + 7/(3x² - 26x - 9)

Question 11. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_{x → ∞} 1/(5x + 7)

Question 12. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_{x → ∞} 9 cos(x)

Question 13. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_{x → ∞} tan^-1(x⁴ - x⁸)

Question 14. Find an equation of the line passing through the given points

(1, 1), (8, - 3/4)

Sketch the line

Question 15. Watch the video below then answer the question

The slope of the tangent line at the point x = a of the function f(x) is

m = lim_{h → 0} (f(a + h) - f(a))/h

True

False

Question 16. Watch the video below then answer the question

The derivative of a function at a point is the slope of the tangent line at that point.

True

False

Question 17. Differentiate the function.

f(x) = 2⁷⁰

f '(x) =

Question 18. Differentiate the function.

f(x) = 5.3x + 2.2

f'(x) =

Question 19. Differentiate the function.

g(x) = 1/6x² - 5x + 13

g'(x) =

Question 20. Find an equation of the tangent line to the curve at the given point.

y = 3x³ - x² + 3, (1, 5)

y =

Question 21. Find an equation of the tangent line to the curve at the given point.

y = 7e^x + x, (0, 7)

Question 22. Find f '(x).

f(x) = x⁵ - 5x³ + x - 1

Compare the graphs of f and f ' and use them to explain why your answer is reasonable.

f '(x) = 0 when f .
f ' is positive when f .
f ' is negative when f

Question 23. The equation of motion of a particle is s = t³ - 12t, where s is in meters and t is in seconds. (Assume t ≥ 0.)

(a) Find the velocity and acceleration as functions of t.

(b) Find the acceleration after 2 s.

(c) Find the acceleration when the velocity is 0.

Question 24. Find the points on the curve y = 2x³ + 3x² - 12x + 6 where the tangent line is horizontal.

(x, y) = (smaller x-value)
(x, y) = (larger x-value)

Question 25. Differentiate.

g(x) = (1 + 8x)/(3 - 2x)

g'(x) =

Question 26. Find an equation of the tangent line to the given curve at the specified point.

y = (x² - 1)/(x² + x + 1), (1, 0)

y =

Question 27. Differentiate.

f(x) = x² sin(x)

f '(x) =

Question 28. Find an equation of the tangent line to the curve at the given point.
y = 8e^x cos(x), (0, 8)

y =

Question 29. If f(x) = 3 sec(x) - 4x, find f '(x).

f '(x) =

Question 30. Find the derivative of the function.

y = e^tan(θ)

Question 31. Find the derivative of the function.

F(x) = (5x⁶ + 8x³)⁴

Question 32. Find the derivative of the function.

f(x) = √(5x + 3)

Question 33.

Find the derivative of the function.

f(t) = 3t sin(Πt)

Question 34. Find an equation of the tangent line to the curve at the given point.

y = sin(sin(x)), (3Π, 0)

Question 35. Find dy/dx by implicit differentiation.

x² - 6xy + y² = 6

y' =

Question 36. Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

y sin(16x) = x cos(2y), (Π/2, Π/4)

y =

Question 37. Differentiate the function.

f(x) = 7x ln(6x) - 7x

f '(x) =

Question 38.

Find an equation of the tangent line to the curve at the given point.

y = ln(x² - 4x + 1), (4, 0)
y =

Question 39.

A cylindrical tank with radius 5 m is being filled with water at a rate of 4 m³/min. How fast is the height of the water increasing?

m/min

Question 40.

A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 35 ft from the pole?

ft/s

Question 41.

Find the numerical value of each expression. (Round your answers to five decimal places.)

(a) sinh(0)

(b) cosh(0)

Question 42.

Find the numerical value of each expression. (Round your answers to five decimal places.)

(a) sinh(1)

(b) sinh^-1(1)

Question 43.

Find the derivative.

f(x) = e^x cosh(x)
f '(x) =

Question 44.

A telephone line hangs between two poles 14 m apart in the shape of the catenary y = 19 cosh(x/19) - 14, where x and y are measured in meters.

(a) Find the slope of this curve where it meets the right pole. (Round your answer to four decimal places.)

(b) Find the angle θ between the line and the pole. (Round your answer to two decimal places.)

Question 45.

(a) Estimate the area under the graph of f(x) = 2/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.)

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?

underestimate
overestimate

(b) Repeat part (a) using left endpoints. (Round your answer to four decimal places.)

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?

underestimate
overestimate

Question 46.

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.

₀∫⁴⁰ sin(√x) dx, n = 4

Question 47.

Use the form of the definition of the integral given in the theorem to evaluate the integral.

₂∫⁵ (16 - 8x) dx

Question 48.

Evaluate the integral.

₇∫⁹(x2 + 2x - 5) dx

Question 49.

Evaluate the integral.

₂∫⁵ √x dx

Question 50.

Evaluate the integral.

_Π/6∫^Π sin(θ) dθ

Question 51.

Evaluate the integral.

₁∫² (v⁵ + 3v⁶)/v⁴ dv

Question 52.

Evaluate the integral.

_1/√3∫^√3 3/(1 + x²) dx

Question 53.

Sketch the region enclosed by the given curves. (A graphing calculator is recommended.)

y = √x, y = 0, x = 4

Calculate its area.

Question 54.

Find the general indefinite integral. (Use C for the constant of integration.)

∫ (u + 6)(2u + 3) du

Question 55.

Find the general indefinite integral. (Use C for the constant of integration.)

∫ 5sin(x) + 2sinh(x)) dx

Question 56.

Find the general indefinite integral. (Use C for the constant of integration.)

∫ 5.sin(2x)/sin(x) dx

Question 57.

Evaluate the integral by making the given substitution. (Use C for the constant of integration.)

∫ cos(4x)dx, u = 4x

Question 58.

Evaluate the integral by making the given substitution. (Use C for the constant of integration. Remember to use absolute values where appropriate.)

∫ x³/(x⁴ -6).dx, u = x⁴ - 6

Question 59.

Evaluate the indefinite integral. (Use C for the constant of integration.)

∫ x√(5 - x²).dx

Question 60.

Evaluate the indefinite integral. (Use C for the constant of integration.)

∫ sec²(θ) tan⁸(θ) dθ

Question 61.

Find the area of the shaded region.

Question 62.

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.

y = e^x, y = x² - 1, x = -1, x = 1

Find the area of the region.

Question 63.

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = x + 1, y = 0, x = 0, x = 8; about the x-axis

V =

Sketch the region.

Sketch the solid, and a typical disk or washer.

Question 64.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y = 2 ³√x, y = 0, x = 1

Question 65.

How much work is done when a hoist lifts a 230-kg rock to a height of 2 m? (Use 9.8 m/s² for the acceleration due to gravity.)

Question 66.

A variable force of 4x^-2 pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from x = 1 ft to x = 10 ft. (Round your answer to two decimal places.)

Question 67.

Find the average value f_ave of the function f on the given interval.

f(x) = 3x² + 4x, [-1, 4]

fave =