Math 151 final Exam Review
J. Lewis
1. All of problem 1 refers to the line x - 3y = 5.
a) Find a vector, v, that is perpendicular to the line.
b) Verify that the point Q(2, -1 ) is on the line. Find the distance from the point P( 3, 2 ) to the line.
2. Find the scalar component and the vector component of <7, 4> in the direction of <2, -3>.
What is the angle between these two vectors?
 x3  9x
 2
x  x  6
3. f ( x )   6 x  2
x

2


x3
3 x5
5 x
a) At what values of x is f not continuous and why is f not continuous there?
b) At what values of x is f not differentiable and why not?
4. f ( x )  ( x
2
 4 x  3)
1 3
At what values of x is f not differentiable? What is seen in the graph at
these x values?
5. For all of problem 2, f ( x ) 
x
2
x 1
x  1
a) Find the equation of the tangent line to f(x) at (1, f(1) ).
b) Find the tangent line approximation (linear approximation ) to f( 1.1 ) . Do not round.
6. a) Find the linear approximation to
5
31 . 5 .
b) Find the linear approximation (linearization) of
1  t for t near 0.
 | x | 4

7. For problem 3, f ( x )   2 x  3
 x 2  4 x  12

x 1
1 x  3
3 x
a) Verify that f is continuous. It is understood that lines and quadratics are continuous.
b) Find all values of x at which f is not differentiable. Justify your answer, either with a graph or some
other means.
8. For problem 4, f ( x )  2 arcsin( x )  ( x 3  1) 2
1  x  1
a) Find f'(x).
b) For g ( x )  f
1
( x ) , find g'(1).
9. Simplify each expression.
3
a ) sec(arcsin ( ))
5
b ) sec(arcsin ( 2 x ))
t
c ) tan( 2 arcsin x )) d ) sin(arctan ( ))
3
10. A blimp is traveling at the constant height of 40 feet with constant speed 5 feet per second, always
in the same direction. A boy on the ground watches the blimp move away from him. What is the rate of
change of the angle between the horizontal and a line from the boy to the blimp at the instant that the
distance between them is 50 feet.
11. A person rows a boat at 3 mph and walks at 4 miles per hour. He is on an island that is 8 miles from
shore. He wants to reach a cabin that is 12 miles down shore. At what point on shore should he tie his
rowboat and walk to minimize his travel time?
12. A curve is given parametrically by x ( t )  ( t  3 ) ( t  4 )
2
y (t )  (t  3) e .
2
t
Find the points on the curve where the tangent line is horizontal and where it is vertical.
x2
13. Evaluate a) lim 

x  x  2 
lim
c)
x  
7  3e
x
2  5e
x
x
lim
x 
d)
 x 
b) lim 

x  0 x  2 
7  3e
x
2  5e
x
x
14. Evaluate each limit.
a)
lim
x 0
sin  2 x 
tan 5 x 
2
2 1 
b ) lim x sin  
x 
x
15. The velocity of an object is given by v ( t )  ( t  5 )( 8  t )  40 miles per hour. (t is in hours)
a) Find the acceleration, a(t).
b) When is the object traveling the fastest?
c) Find the distance traveled in the first 2 hours.
d) At what value of t does the graph of D(t),the distance traveled in t hours, have an inflection point.
16. Find the derivative of each function.
a ) f ( x )  2 xe
3x
2
b) f ( x) 
x3
( x  1)
2
17. Find the nth derivative of a ) f ( x )  e
3x
4
2
b ) f ( x )  2 xe
18. Find the 43rd derivative of f ( x )  sin x cos x .
25
19. Evaluate a )
 5  3 k 
k 1
20
b)
 4k
k 8
c ) f ( x )  tan ( 5 x )
x
c) f (x)  2
x
20.Find the right hand, left hand and midpoint Riemann sums using n equal subintervals for the given
function, interval and n. Sketch the function and the rectangles.
a) f ( x )  x
b) f ( x )  8
 9 on the interval [ 2,4 ],n=4
2
2x
on [0,1],
Evaluate lim R n .
n 
n 3
21. Find each antiderivative.
a)  sin x dx
b)  2 sec
2
( x  1)
c) 
d) 
x
xe
x  3 cos xdx
3
x dx
 5x  1
3
dx
x
sec t (sec t  tan t ) dt
e) 
5x  6
2
f) 
x 1
2
dx
22. Evaluate each definite integral.
3
a)

9 x
2
3
dx
2
0
3
b)
 4e 
0
x
5
x 1
x 1

1
dx
2
c)
1
1 x
2
dx