«
»

Section 3.5

By cssp

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) f'(x) = x^2\cos(x) + 2x\sin(x)

(c) g'(t) = -6t\sin(2t) + 3\cos(2t)

(e) f'(t) = -4\sin(3)t\sin(4t) + 3\cos(3t)\cos(4t)

(f) g'(z) = 12\sin^2(4z)\cos(4z)

2. (a) \displaystyle{\frac{dy}{dx} = \frac{2x\cos(2x) - \sin(2x)}{x^2}}

(c) \dfrac{dx}{dt} = 8t\cos(4t^2 + 1)

(e) \dfrac{dz}{dt} = 2\sec(2t)\tan(2t)

(g) \dfrac{dy}{dx} = -2x^2\csc(2x)\cot(2x) + 2x\csc(2x)

3. \dfrac{d}{dx}(\sin^2(2x)\cos^2(3x) = -6\sin^2(2x)\cos(3x)\sin(3x)
+ 4\cos^2(3x)\sin(2x)\cos(2x)

(c) \dfrac{d}{dq}\sec^3(q^2) = 6q\sec^3(q^2)\tan(q^2)

(e) \displaystyle{\frac{d}{dz}\sqrt{1 + \sin^2(z)} = \frac{\sin(z)\cos(z)}{\sqrt{1 + \sin^2(z)}}}

5. T(t) = 1

7. (a) S(x) = \dfrac{1}{2}x + 1

(b) T(x) = 4x

(c) U(x) = 2x + 1

(e) h = f \circ g and U = S \circ T

8. (a) \displaystyle{\lim_{x \to 0}\frac{\sin(2x)}{x} = 2}

(c) \displaystyle{\lim_{x \to 0}\frac{\tan(x)}{x} = 1}

(e) \displaystyle{\lim_{x \to 0}\frac{\sin^2(x)}{x} = 0}

(g) \displaystyle{\lim_{t \to 0}\frac{\sin^2(3t)}{t^2} = 9}

9. (a) f is O(h) , but not o(h) .

(c) g is O(h) , but not o(h) .

(e) f is o(h) , and so also O(h) .

11. (a) D(0.00001) = 4.000010000027032 to 15 decimal places
D_1(0.00001) = 4.000000000026205 to 15 decimal places
D_2(0.00001) = 3.999999999848569 to 15 decimal places
f'(2) = 2

(c) D(0.00001) = 0.999999999983333 to 15 decimal places
D_1(0.00001) = 0.999999999983333 to 15 decimal places
D_2(0.00001) = 1.000000000000000 to 15 decimal places
f'(0) = 1

This entry was posted on 13 July 13 2007 at 3:33 pm and is filed under Section 3.5. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Leave a Reply