Name: College ID: Thomas Edison State College Calculus I (MAT-231) Section no.: Semester and year: Written Assignment 3 Answer all assigned exercises, and show all work. Each exercise is worth 4 points. 1. Use Definition 1.1 (See Section 2.1) to find an equation of the tangent line to y = f ( x) at x =a . Graph y = f ( x) and the tangent line to verify that you have the correct equation. f ( x) = x +3, a =1 2. Use the position function s (in meters) to find the velocity at time t =a seconds. s (t ) =4 / t , (a) a =2; (b) a =4 3. Compute the derivative function f using Definition 2.1 or Definition 2.2 (See Section 2.2). f ( x) =x 2 - 2 x +1 4. Compute the derivative function f using Definition 2.1 or Definition 2.2 (See Section 2.2). f ( x) = 2 2x - 1 WA 3, p. 1 5. Differentiate the function. h( x) =12 x - x 2 - 3 x2 6. Differentiate the function. f (t ) =3t p - 2t 1.3 7. Differentiate the function. 4 x 2 - x +3 f ( x) = x 8. Compute the indicated derivative. 4 f (t ) for f (t ) =4t 2 - 12 + 2 t 9. Use the given position function to find the velocity and acceleration functions. s (t ) =- 4.9t 2 +12t - 3 10. Find an equation of the tangent line to y = f ( x) at x =a . f ( x) =x 2 - 2 x +1, a =2 11. Find the derivative. 3 f ( x) =( x 3/2 - 4 x) 4 - 2 +2 x x 12. Find the derivative. f ( x) = x2 - 2x x 2 +5 x 13. Find an equation of the tangent line to the graph of y = f ( x) at x =a . f ( x) =( x 3 +x +1)(3 x 2 +2 x - 1), a =1 14. Differentiate each function. a) b) f (t ) =(t 4 +2) t 2 +1 f (t ) = t (t 4/3 +3) 15. Differentiate each function. a) h( w) = ( w3 +4)5 8 WA 3, p. 3 h( w) = 8 ( w +4)5 3 b) 16. Find an equation of the tangent line to the graph of y = f ( x) at x =a . f ( x) = 6 , a =- 2 x +4 2 17. Find the derivative. f (t ) = cos 5t sec 5t 18. Find the derivative. f ( w) =w2 sec 2 3w 19. Find the derivative. f ( x) =4sin 2 3 x +4 cos 2 3 x 20. Differentiate each function. a) f ( x) = ln x x g (t ) = ln t t b) 21. Differentiate each function. a) f ( x) = 3 e 2 x x 3 b) f ( w) = 3 e 2 w +w3 22. Find the derivative y ( x ) implicitly. 3x +y3 - 4y =10 x 2 x +2 23. Find an equation of the tangent line at the given point. If you have a CAS that will graph implicit curves, sketch the curve and the tangent line. x 3 y 2 =- 2 xy - 3 at (- 1, - 3) 24. Suppose that you are blowing up a balloon by adding air at the rate of 1 ft3/s. If the 3 4 balloon maintains a spherical shape, the volume and radius are related by V = 3 p r . Compare the rate at which the radius is changing when r = 0.01 ft versus when r = 0.1 ft. Discuss how this matches the experience of a person blowing up a balloon. WA 3, p. 5 25. Sand is dumped such that the shape of the sandpile remains a cone with height equal to twice the radius. (a) If the sand is dumped at the constant rate of 20 ft3/s, find the rate at which the radius is increasing when the height reaches 6 feet. (b) Repeat for a sandpile for which the edge of the sandpile forms an angle of 45o with the horizontal