Calculus 1  Final  Exam    -   Math 220   Fall, 2006

 

 

1.

 

 

Sketch the parabola given by   y = x2 + 6x – 1.  Find its vertex and x and y intercepts.

 

Answers:    y-intercept = -1            x-intercepts at  -3 ± √10

 

 

2.

 

 

Use the definition of the derivative to find  df/dx for each  x for the function, f(x) = 2x – 3.

 

 

3.

 

Write the equation of the line tangent to the curve  y = 4 x 2/3   at  (1,4).

 

Answer:  y = (8/3) x  +  4/3

 

 

 

4.

 

Starting from  xo = 1, find  x1  and  x2  using Newton’s method to approximate

solution of the equation   f(x) =  2 ln x  - 2 +  x.

 

Answers:   x1  = 4/3      x2  = 4/3 – (2/5) [ 2 ln(4/3) – 2/3 ]

 

 

 

5.

 

 

Find the intervals of convexity (up, down) of the function  f(x) = 2x3 – 9x2 – 102 x + 5.

 

Answers:   Concave up for  x > 3/2      Concave down for  x < 3/2

 

 

6.

 

 

Find the derivative of  y = arcos x.

 

Answer:  dy/dx  =  -1 / √ (1 – x2 )

 

 

 

 

 

7.

 

 

 

Use the method of cross-sections (disks) to find the volume of the solid generated by

rotating around the y-axis the region bounded by the curves x= 0, y = x, and

y = 3 – 2x.

 

Answer:  V = π.

 

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