Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Review Questions 2

MAT1300X, Summer 2022

1.  Find the local/global maxima/minima, if any, of the functionf (x) =  x4 – x3 − 2x2 defined on a closed interval −2  x  5.

2.  Suppose a function y =f (x) defined and continuous for all x, except x = 0, satisfies the following conditions:

(i) f '(x) < 0 when x < 2;f ' (2) = 0;f '(x) > 0 when 2 < x < 0 or x > 0;f '(0) is not defined.

(ii) f "(x) > 0 when x < 0 or x > 2;f "(x) < 0 when 0 < x < 2;f "(2) = 0;f "(0) is not defined.

(iii)   lim f (x) = , lim f (x) = , lim f (x) = lim f (x) =  .

x0                                  x0+                                                          x                        x

(iv) f (−2) =f (2) = 0.

(a)  Determine the interval(s) wheref (x) is increasing, and the interval(s) wheref (x) is decreasing.

(b)  Find the local maxima and local minima, if any.

(c)  Determine the interval(s) where the graph off (x) is concave up, and the interval(s) where the graph off (x) is concave down.

(d)  Find the inflection point(s) off (x), if any.

(e)  What are the horizontal/vertical asymptotes of the graph off (x), if any?

(f)  Sketch the graph of the function.

e   

3.  Consider functionf (x) =

(a)  Find the first and the second derivatives of this function.

(b)  Find critical numbers of this function.

(c)  For which values of x is this function increasing / decreasing?  Find all local max / min of this function, if any.

(d)  For which values of x is the graph of this function concave up / down?  Find all inflection points, if any.

(e)  Find all vertical/ horizontal asymptotes, if any.

(f)  Sketch the graph of this function.

4.  Suppose a farmer wants to use fence to enclose a rectangular region of area 1500m2, and separate this region into three parts by inner fence as in the following figure:

inner fence

 

 

 

 

 

 

 

ou

 

 

 

ter fence

                        x                        

y

The outer fence costs $60 per meter, and the inner fence costs $40 per meter.  Let the width of the region be x, and let the depth of the region be y.  Find x and y so that the cost of fence is    minimized.  Justify what you found is a global minimum.

5.  Suppose the demand function of a product is p = 2 +  , 0 < q  4, where p is the unit price, and q is the quantity of this product that can be sold every day with price p.  The cost function of this product is C(q) = 5q.  Find the quantity q that maximizes the profit.  Justify that what you     found is the global maximum.

6.  Find an approximation of  using the tangent line approximation of functionf (x) =   at x = 5.

7.  Suppose the demand function of a product is q =   , 0  p  5.

(a)  Find the elasticity of demand.

(b)  For which values ofp is the demand elastic?

(c)  If the current price is p = 3.5, to increase the revenue, shall the producer increase the price to sell fewer products, or decrease the price to sell more products?

8.  Find the derivative of the implicit function y =f (x) defined by the equation xy + y3  x3 = 3 near a point (x, y) on the graph of this equation.

9.  Find the equation of the tangent line of the graph of equation  + 2x + y2 = 3 at the point (2, −1).

10.  Use n = 4 to find approximations of the integral j1  x2dx with L(4), R(4), and T(4).

11.  If j1  f (x)dx = 7, j1  f (x)dx = 13, and j4 f (x)dx = 2, find j4  f (x)dx .

12.  If j1  f (x)dx = 7 and j4 f (x)dx = 4, find j1  (2 + 3  - 5f (x))dx .

1 + 2