1. Differentiate the following functions as indicated:

a)  h(x) = 3  x2   + ln | 5 + 4x |  sin(几 x) , find h ,(0)

c)   f (x) = (x 5   4x 1.5 )( 3x2  + 1) , find f ,(1)

d)  6(x) = c卜L一(9)Zx   find 6′(−)

2.   Given the equation  x cos y 一 sin y = 5x   find    by implicit differentiation.

3.   Find the points on the graph of the function f (x) = 一2xe一x   where the tangent line in horizontal.

4. Find all the points on the graph of  f (x) =  x 3   3x2  + 5x  4 where the tangent line is parallel to the line 3x + y = 5 .

5. Use the limit definition of the derivative to find f ,(3) when  f (x) = 1    .

6.   Graph the function f (x) =〈(52(-) 2x    if  x   the definition of continuity to

7. Find the value(s) of c and d such that the function  f (x) =〈 continuous and differentiable at x=2.

if  x  2

is

8. The production of a certain commodity is increasing at a rate of 80 units per month. The

demand and cost functions are respectively: p = 240  0. 15x    and

C(x) = 9600 + 100x + 0.05x2 .

a) Find the marginal cost of producing 600 units. Interpret your result .

b) Find the rate of change of the profit with respect to time if the production level is 600 units.

9.   Find the first partial derivatives of the function f (x, y) =  3xy + 6x + ln(x + y) + 2y 3 at the point (0, 1). 

Differential Calculus

10. An open rectangular box having a surface area of 300 cm2  is to be constructed from a tin sheet. Find the dimensions of the box if the volume of the box is to be as large as  possible. What is the maximum volume? 

Differential Calculus

11. Find the relative extrema of the function f (x, y) = xy +  +  .

12. Evaluate the following limits if they exist.  If they do not exist,  indicate why not.

a) lx                                                               b) lix 

c) xl(x 3   e x )

d) xl 

13. Use the linear approximation to approximate 632 / 3 .  Use the second derivative test to check if your approximation is too big or too small compared to the real value.

14. Given  f (x) = x3 (x  5)2 ,  f ' (x) = 5x2 (x  5)(x  3) ,  and

f "(x) = 10x(2x2   12x + 15) , sketch a complete graph off.  Be sure to clearly indicate all intercepts, relative extrema, concavity, and inflection points.

15. Consider the function f (x) =   and its derivatives f  (x) =   and f (x) =  .  Sketch the graph of the function.

Differential Calculus

16. A balloon is rising at a constant speed 4m/sec. A boy is cycling along a straight road at a speed of 8m/sec. When he passes under the balloon, it is 36 metres above him.   How fast is the distance between the boy and balloon increasing 3 seconds later?

17. A closed rectangular box of volume 20 m3  is to be constructed with a square base of    width x. The material for the top costs $5 per square metre whereas the material for     the remaining sides costs $2 per square metre. Find the cost of the box as a function of the width of the base. Find the dimensions of the cheapest box. 

Differential Calculus

18. Graph the functions y = ex   and y =   + 3 . Use Newton’s Method to approximate the x-value of intersection of the two graphs.