Reference no: EM131684070

Math Review for Calculus Homework

Q1. Convert from degrees to radians - 330^o.

Q2. Convert from radians to degrees - 13π/18.

Q3. Find the exact value - tan π/3.

Q4. Find the exact value - sin 4π/3.

Q5. If sin(x) = 1/3 and sec(y) = 5/3, where x and y lie between 0 and π/2, evaluate sin(x + y).

Q6. Find all values of x such that sin(2x) = sin(x) and 0 ≤ x ≤ 2π. (Enter your answers as a comma-separated list.)

Q7. Sketch the graph of the function y = 2 + sin(2x) without using a calculator.

Q8. 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)

Q9. 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?

Q10. Find the horizontal and vertical asymptotes of the curve.

y = (x² + 7)/(3x² - 26x - 9)

Q11. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_x→∞ 1/5x + 7

Q12. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_x→∞9 cos(x)

Q13. Find the limit, if it exists. (If an answer does not exist, enter DNE.)

lim_x→∞ tan^-1(x⁴ - x⁸)

Q14. Find an equation of the line passing through the given points.

(1, 1),  (8, -3/4)

Sketch the line.

Q15. 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    

Q16. The derivative of a function at a point is the slope of the tangent line at that point.

True

False    

Q17. Differentiate the function - f(x) = 2⁷⁰.

Q18. Differentiate the function - f(x) = 5.3x + 2.2.

Q19. Differentiate the function - g(x) =  (1/6)x² - 5x + 13.

Q20. Find an equation of the tangent line to the curve at the given point.

y = 3x³ - x²  + 3,  (1, 5)

Q21. Find an equation of the tangent line to the curve at the given point.

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

Q22. 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.

Q23. 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.

Q24. Find the points on the curve y = 2x³ + 3x² - 12x + 6 where the tangent line is horizontal.

Q25. Differentiate - g(x) = 1 + 8x/3 - 2x.

Q26. Find an equation of the tangent line to the given curve at the specified point.

y = (x² - 1)/(x² + x + 1),  (1, 0)

Q27. Differentiate - f(x) = x² sin(x).

Q28. Find an equation of the tangent line to the curve at the given point.

y = 8e^x cos(x),    (0, 8)

Q29. If f(x) = 3 sec(x) - 4x, find f'(x).

Q30. Find the derivative of the function - y = e^tan(θ).

Q31. Find the derivative of the function - F(x) = (5x⁶ + 8x³)⁴.

Q32. Find the derivative of the function - f(x) = √(5x + 3).

Q33. Find the derivative of the function - f(t) = 3t sin(πt).

Q34. Find an equation of the tangent line to the curve at the given point.

y = sin(sin(x)),  (3π, 0)

Q35. Find dy/dx by implicit differentiation.

x² - 6xy + y² = 6

Q36. 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)

Q37. Differentiate the function.

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

Q38. Find an equation of the tangent line to the curve at the given point.

y = ln(x² - 4x + 1),    (4, 0)

Q39. 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?

Q40. 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?

Q41. Find the numerical value of each expression. (Round your answers to five decimal places.)

(a) sinh(0)   

(b) cosh(0)   

Q42. Find the numerical value of each expression. (Round your answers to five decimal places.)

(a) sinh(1)

(b) sinh^-1(1)

Q43. Find the derivative.

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

Q44. A telephone line hangs between two poles 14 m apart in the shape of the catenary  y = 19cosh(x/19) - 14, where x and y are measured in meters.

(a) Find the slope of this curve where it meets the right pole.

(b) Find the angle θ between the line and the pole.

Q45. (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.

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?

• underestimate
• overestimate

(b) Repeat part (a) using left endpoints.

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?

• underestimate
• overestimate

Q46. 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

Q47. Use the form of the definition of the integral given in the theorem to evaluate the integral.

₂∫⁵ (16 - 8x) dx

Q48. Evaluate the integral - ₇∫⁹ (x² + 2x - 5) dx.

Q49. Evaluate the integral - ₁∫⁴ √x dx.

Q50. Evaluate the integral - _π/6∫^πsin(θ) dθ.

Q51. Evaluate the integral - ₁∫² (v⁵+3v⁶)/v⁴ dv.

Q52. Evaluate the integral - _1/√3∫^√3 3/(1+x²) dx.

Q53. Sketch the region enclosed by the given curves. (A graphing calculator is recommended.)

y =  √x, y = 0, x = 4

Q54. Find the general indefinite integral. (Use C for the constant of integration.)

∫(u + 6)(2u + 3) du

Q55. Find the general indefinite integral. (Use C for the constant of integration.)

∫(5 sin(x) + 2 sinh(x)) dx

Q56. Find the general indefinite integral. (Use C for the constant of integration.)

∫5(sin(2x)/sin(x))dx

Q57. Evaluate the integral by making the given substitution. (Use C for the constant of integration.)

∫cos(4x) dx, u = 4x

Q58. 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

Q59. Evaluate the indefinite integral. (Use C for the constant of integration.)

∫x√(5 - x²)dx

Q60. Evaluate the indefinite integral. (Use C for the constant of integration.)

∫sec²(θ) tan⁸(θ) dθ

Q61. Find the area of the shaded region.

Q62. 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

Q63. 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

Sketch the region.

Sketch the solid, and a typical disk or washer.

Q64. 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 = 2³√x, y = 0,  x = 1

Q65. How much work is done when a hoist lifts a 230-kg rock to a height of 2m? (Use 9.8 m/s² for the acceleration due to gravity.)

Q66. 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.)

Q67. Find the average value f_ave of the function f on the given interval.

f(x) = 3x² + 4x,    [-1, 4]