Worksheet 6 - Tiezheng QIAN PDF

Preview

Summary

Tiezheng QIAN...

Description

MATH1013 Calculus IB, 2018-19 Fall semester Tutorial Worksheet 6: Chain rule/implicit differentiation and rates of change (T10A, T10B) Name:

ID No.:

Tutorial Section:

Your tutor would demonstrate some exemplary questions. You would be asked to work out some problems yourselves. (Solution of this worksheet will be available at the course website after all the tutorials of that week) √ 1. (Demonstration)(Stewart: Ex. 3.4) Find the derivative of Q. 7 (x4 + 3x2 − 2)5 , Q. 12 3 1 + tan t. 2. (Demonstration) (Stewart: Ex. 3.4) Find the derivative of Q. 35 y = sin(sin(sin x)).

 1 − cos 2x 4 1 + cos 2x

, Q. 40 y =

3. (Demonstration) (Stewart: Ex. 3.4, Q. 61) If F (x) = f (g(x), where f (−2) = 8, f ′ (−2) = 4, f ′ (5) = 3, g (5) = −2, and g ′ (5) = 6, then Find F ′ (5). 4. (Demonstration) (Stewart: Ex. 3.5, Q. 27) Find tangent line equation(s) to x2 + xy + y2 = 3 at x = 1.

5. (Demonstration) (Stewart: Ex. 3.6) Find the derivative of Q. 7 f (x) = log10 (x3 + 1), and Q. 17 f (x) = tan(tan(ax + b)). 6. (Demonstration) The √ lateral surface area of a cone of radius r and height h (the surface area excluding the base) is A = πr r2 + h2 . a. Find dr/dh for a cone with a lateral surface area of A = 1500π. b. Evaluate this derivative when r = 30 and h = 40. 7. (Class work) Find the derivative of (x3 + 3x− 2)8 ,

√ 5 1 + tan t

Answer 8. (Class work) Find the derivative of y =

Answer

 1 − cos 2x 3 1 + cos 2x

, and y = cos(cos(cos x)).

9. (Class work) Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g (x) = f (x2 ) and the tangent line of f at the point (1, 4) is y = 3x + 1. Determine each of (a) g (1), (b) g ′ (x), (c) g ′ (1), (d) an equation of the tangent line to graph of g when x = 1.

Answer 10. (Class work) (Stewart: Ex. 2.6, Q. 28) Find the tangent line equation to x2 + 2xy −2 +x = 2 at (1, 2).

Answer 11. (Class work) (Stewart: Ex. 3.6) Find the derivative of Q. 6 f (x) = ln(sin2 x), and Q. 16 f (t) = ln |1 + t − t2 |.

Answer 12. (Class work) The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V = π 2 (b + a)(b − a)2 /4. a. Find db/da for a torus with a volume of 64π 2 . b. Evaluate this derivative when a = 6 and b = 10.

Answer...