Section 3.5

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

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

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

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

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

(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) $T(x) = 4x$

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

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

(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 $o(h)$ .

(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
$f'(0) = 1$

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Difference Equations to Differential Equations

Section 3.5

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