McLoughlin
P-Final Math 192
Name:
1.) Let f (x) = x3 + 5x2 − 6. Find the following:
a.) (4 points) The equation of the secant line through (1, f (1)) and (2, f (2));
b.) (4 points) The equation of the tangent line through (1, f (1)).
2.) (4 points each) Find the derivative of each function.
a.) f (x) =
x3 +5x2 −6
3x2 −7x+1
b.) f (x) = (x3 + 5x2 − 6)−12
3.) (8 points) How long will it take a population that is growing at 6% per year to double in
size?
4.) (8 points) A ladder 20 feet long is leaning against a wall of a house. The top of the ladder
starts to slide down the wall at a rate of 2ft/sec. How fast is the bottom of the ladder moving along
the ground when the top of the ladder is 10 feet from the ground?
5.)
a.)
b.)
c.)
d.)
e.)
For the given function find the following:
(3 points each) Intervals where f is increasing and decreasing;
(1 points each) All local extrema;
(3 points each) Intervals where f is concave up and concave down;
(1 points each) Points of inflection;
(3 points each) Sketch the graph of f.
f (x) = 7x3 + 36x2 − 165x + 3
6.) (7 points)Find the x- and y-intercepts, and any vertical or horizontal asymptotes.
f (x) =
4x2 −9
(x−5)(x+2)(2x−7)
7.) (10 points) A rectangular storage container with a closed top is to have a volume of 50m3 .
The width of the base is four times the height. Material for the base costs $12 per square meter.
Material for the top costs $5 per square meter. Material for the sides cost $7 per square meter. Find
the cost of materials for making the cheapest container.
8.)(7 points)Use the definition of the definite integral with the given value of n to approximate
the definite integral.
R 12
2
(x3 −
3
x
− 1)dx,
n=5
9.) (3.5 points each) Integrate.
a.)
R 12
b.)
R
(3x2 − 9)(x3 − 9x − 5)−12 dx
c.)
R
−12xe−6x +15x2 −4
dx
(e−6x2 +5x3 −4x−2)5
d.)
R
1
x(lnx)7 dx
2
(x3 −
3
x
− 1)dx
2
10.)(5 points each) The velocity function in meters per second is given for an object moving along
a line. Find:
a.) the displacement on [0, 7];
b.) the distance traveled by the object.
v(t) = (5t − 6)2 − 16 on [0, 7]
11.)(9 points) Evaluate using integration by parts.
R
(8x2 − 2x + 1)ex dx
2