1. [T] Find expressions for and Use a calculator to graph these functions and ensure your expression is correct.
2. From the definitions of and find their antiderivatives.
3. Show that and satisfy
4. Use the quotient rule to verify that
5. Derive from the definition.
6. Take the derivative of the previous expression to find an expression for
7. Prove by changing the expression to exponentials.
8. Take the derivative of the previous expression to find an expression for
For the following exercises (9-18), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.
9. [T]
10. [T]
11. [T]
12. [T]
13. [T]
14. [T]
15. [T]
16. [T]
17. [T]
18. [T]
For the following exercises (19-29), find the antiderivatives for the given functions.
19.
20.
21.
23.
24.
25.
26.
27.
28.
29.
For the following exercises (30-36), find the derivatives for the functions.
30.
31.
32.
33.
34.
35.
36.
For the following exercises (37-43), find the antiderivatives for the functions.
37.
38.
39.
40.
41.
42.
43.
For the following exercises (44-46), use the fact that a falling body with friction equal to velocity squared obeys the equation
44. Show that satisfies this equation.
45. Derive the previous expression for by integrating
46. [T] Estimate how far a body has fallen in 12 seconds by finding the area underneath the curve of
For the following exercises (47-49), use this scenario: A cable hanging under its own weight has a slope that satisfies The constant is the ratio of cable density to tension.
47. Show that satisfies this equation.
48. Integrate to find the cable height if
49. Sketch the cable and determine how far down it sags at
For the following exercises (50-53), solve each problem.
50. [T] A chain hangs from two posts 2 m apart to form a catenary described by the equation Find the slope of the catenary at the left fence post.
51. [T] A chain hangs from two posts four meters apart to form a catenary described by the equation Find the total length of the catenary (arc length).
52. [T] A high-voltage power line is a catenary described by Find the ratio of the area under the catenary to its arc length. What do you notice?
53. A telephone line is a catenary described by Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?
54. Prove the formula for the derivative of by differentiating (Hint: Use hyperbolic trigonometric identities.)
55. Prove the formula for the derivative of by differentiating
(Hint: Use hyperbolic trigonometric identities.)
56. Prove the formula for the derivative of by differentiating (Hint: Use hyperbolic trigonometric identities.)
57. Prove that
58. Prove the expression for Multiply by and solve for Does your expression match the textbook?
59. Prove the expression for Multiply by and solve for Does your expression match the textbook?
Candela Citations
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction