Avon High School
Name _________________________________________
VU Calculus
UNIT 5 (3.5-3.7, 3.9) EXAM REVIEW
Limits at Infinity/Curve Sketching/Optimization/Differentials
Find the vertical asymptote(s), if any.
x2  x  6
1.) y 
x2  4
Find the horizontal asymptote(s), if any.
7x 1
3 x  1
3.) y  2
4.) y 
6x  7
5 x  4
2.) y 
4
2 x  7 x  15
5.) y 
5 x 2  1
3x
2
6.) y 
5 x
64 x 2  3
The graph of the function f is shown. Sketch the graph of the derivative, f  .
7.)
8.)
Analyze and sketch a graph of the given function.
x
x2
9.) f  x  
10.)
f
x



1  x4
x2  1
11.) f  x   x 9  x 2
Determine the slant asymptote of the graph of f .
12.) f  x  
2 x 2  6 x  3
x 1
The graph of f is shown to the right.
14.) On what interval(s) is f   0 ?
15.) For which value of x is f  zero?
13.) f  x  
x 2  8 x  15
x6
16.) Find two positive numbers whose product is 187 and whose sum is a minimum.
17.) Find the length and width of a rectangle that has an area of 288 square feet and whose perimeter is a
minimum.
18.) Find the point on the graph of the function f  x   x that is closest to the point 18,0 .
19.) Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface
area is 121 meters.
5 x
(see
2
figure). What length and width should the rectangle have so that its area is a
maximum?
20.) A rectangle is bounded by the x - and y -axes and the graph of y 
21.) Find the equation of the tangent line T to the graph of f  x  
23
 23 
at the given point  2,  .
2
x
 4 
22.) Compare dy and  y for y  2 x 2  4 at x  2 with x  dx  0.05 . Give your answers to four decimal
places.
23.) Compare dy and  y for y  x 4  4 at x  0 with x  dx  0.05 . Give your answers to four decimal places.
Find the differential dy of the function.
24.) y   x 2  3x  4
25.) y  4 x5 8
26.) y  x sin  5x 