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Section 3.2

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1. (a) f'(2) = 4

(c) g'(2) = -\dfrac{1}{4}

(e) f'(1) = -\dfrac{1}{2}

2. (a) Best affine approximation: T(x) = 4(x - 2) + 5
Equation of tangent line: y = 4(x - 2) + 5

(c) Best affine approximation: \displaystyle{T(x) = -\frac{1}{4}(x - 2) + \frac{1}{4}}
Equation of tangent line: \displaystyle{y = -\frac{1}{4}(x - 2) + \frac{1}{4}}

(e) Best affine approximation: \displaystyle{T(s) = -\frac{1}{2}(s - 1) + 1}
Equation of tangent line: \displaystyle{y = -\frac{1}{2}(s - 1) + 1}

3. (a) f'(x = 4x

(c) \displaystyle{f'(t) = -\frac{1}{2t^{\frac{3}{2}}}}; f is not differentiable at t = 0

(e) y'(t) = 2t + 4

4. Best affine approximation: \displaystyle{T(t) = -\frac{1}{16}(x - 4) + \frac{1}{2}}
\displaystyle{\frac{1}{\sqrt{3.98}} \approx T(3.9 <img src='http://s.wordpress.com/wp-includes/images/smilies/icon_cool.gif' alt='8)' class='wp-smiley' /> = \frac{401}{800} = 0.50125} to five decimal places

7. T(t) = -9(t - 3) - 15

11. f'(0) = 0

13. g is not differentiable at x = 1 .

14. (a) \dfrac{ds}{dt} = 6t

(c) \displaystyle{\frac{dq}{ds} = 1 + \frac{2}{s^2}}

15. \dfrac{d}{dx}(4x^2) = 8x , \displaystyle{\frac{d}{du}(3\sqrt{u-1}) = \frac{3}{2\sqrt{u-1}}}

16. (a) Average velocity over [0, 1] = 0.2 \text{ meters/second}

(b) v = 10 - 9.8t \text{ meters/second}

(c) v|_{t=1} = 0.2 \text{ meters/second}

(d) v = 0 when t = 1.0204 seconds. v is positive before this time, and negative after.

(e) s|_{t=1.0204} = 105.10 \text{ meters} , which must be the maximum height reached by the object.

(f) \dfrac{dv}{dt} = -9.8 \text{ meters/second}^2

(g) The object is losing velocity at the same rate at all times.

17. \dfrac{dA}{dr} = 2\pi r

18. \dfrac{dV}{dr} = 4\pi r^2

19. \dfrac{dA}{dx} = 2x

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