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1. Evaluate the following limits.

a. lim_x2(1/x-2-12/x³-8)

b lim x-∞2+x/4+2x

c lim –o (1/x1+x-1/x)

2. Use the limit definition of the derivative to show the following.

3 a) If f(x) = ——, then f’(1) = -}

[3] (a) f(z) = 5; then fl) = —4

1

3 b) If f(z) = Vx +1, tl “2) = ——

3 (b) If fle) = VEFI, then f'(2) = 7

3. Find the derivative of the following functions.

3 (a) f(x) = e^x2+x-e2x/x2-1

. r) —e€ __*

Vx? -—1

xl/? + In(2x)

b ©) = ———_—_

4. The curve with equation 2x5 + 5y3 = 7 passes through the point (1, 1).

[2] (a) Use implicit differentiation to find = at (1,1).

[2] (b) Find the equation of the line which is tangent to the curve at (1,1).

[4] 5. Suppose the volume of a sphere is expanding at the rate of 1.5 (m* per minute).

How quickly is its surface area increasing at the moment when its volume is 4a (m*)?

, ; a 4nr®

Hint: For a sphere of radius 7, the volume is given by V = a and the

surface area is given by S = 4zr?.

6. Let f(x) = e7* —e7**,

[1] (a) What is the domain of f .

[1] (b) Find the asymptote(s) of the function, if exists .

[1] (c) Find the x- and y- intercepts of y = f(z) .

[2] (d) Determine lim f(z).

z—+00

[1] (e) On which intervals is f(x) increasing or decreasing ?

[1] (f) On which intervals is f(x) concave up or down ?

 

[3] (g) Use the above parts to sketch the graph of the function. Label all critical points, points of inflections, x- and y- intercepts, local minima and maxima, and asymptotes, if exist.

 

[4] 7. Use the linear approximation to show \/1 + /z = zy

 

Hint: Use f(z) = /1+ 2, with a = 4.

8. Evaluate the following integrals (show all of your works).

[3] (a) f In(x) dx

1

2 3

[3] (b) [ xe” dr

0

[6] 9. Determine the length of the curve y(x) = 42? — 3 In(x) on the interval [1, 2].

[6] 10. A bacterial population starts with 150 bacteria. Its growth rate N(t) varies with respect to time ¢. If the growth rate N(t) equals 300 e” (bacteria/hour), how many bacteria will there be after three hours?