Math 151 Final Exam, Sample
2
1
.
x−2
[6]
1. Use the limit definition of the derivative to find f 0 (x), given that f (x) = √
[5]
2. Use the definition of continuity to determine whether or not f is continuous at x = 0.
(
2
e−2/x
f (x) =
0
if x 6= 0
if x = 0
Math 151 Final Exam, Sample
3. Evaluate the limit, if it exists. If it does not exist, show why.
[3]
2
(a) lim x · cos
x→0
x
[3]
(b) lim−
4
x→2
|x−2|
x2 − 4
[5]
(c) lim+
x→1
1
x−1
log x
3
Math 151 Final Exam, Sample
4
4. Find the derivative of the following functions.
3x
+ tan
x
x−2
[3]
(a) f (x) = 2
[4]
√
(b) f (x) = sin−1 ( 2x + 1) − log3 (6x)
[4]
(c) f (x) = xcos x
Math 151 Final Exam, Sample
[6]
5. A car brakes with a constant deceleration of 16 ft/s2 and continues to travel 200 feet
before stopping. What was the speed of the car when the brakes were first applied?
[4]
6. Use linear approximation to approximate the value of ln(0.8). Is this an underestimate
or an overestimate?
[5]
7. A cup of hot chocolate has temperature 80o C in a room kept at 20o C. After half an
hour the hot chocolate cools to 60o C. What is the temperature of the hot chocolate
after another half hour?
5
Math 151 Final Exam, Sample
[6]
8. A light is at the top of a pole 25 m high and a ball is dropped from the same height at a
point 10 m from the light. How fast is the shadow of the ball moving along the ground
1 second later? Assume the ball falls a · t2 meters in t seconds, where a is a constant.
[7]
9. Consider the triangle AOB shown in the figure where the hypotenuse passes through
the point (2, 5). Find the coordinates of A(0, y) and B(x, 0) such that the area of the
triangle is a minimum. Justify that your answer is a minimum.
A(0, y)
(2, 5)
O
B(x, 0)
6
Math 151 Final Exam, Sample
10. Let f (x) =
7
3(1 − x2 )
6(x3 − 3x + 1)
3(x2 + 1)
0
00
then
f
(x)
=
;
f
(x)
=
x2 − x + 1
(x2 − x + 1)
(x2 − x + 1)2
[4]
(a) Check if the graph of f has any vertical or horizontal asymptotes. Give the equation(s) of the asymptote(s) that you find.
[4]
(b) Find the critical numbers and determine intervals where f is increasing or decreasing
and the points of local extrema.
[6]
(c) Sketch the graph of f and label all points found in (a) and (b).
6
5
4
3
2
1
0
−6 −5 −4 −3 −2 −1 0
−1
1
2
3
4
5
6
−2
−3
−4
−5
[2]
(d) Give points of inflection. Justify your claim.
Math 151 Final Exam, Sample
8
11. Suppose y = 9 − 2x is the tangent line to a curve y = f (x) at the point (2, 5).
[3]
(a) If Newton’s Method is used to locate a root of the equation f (x) = 0 and the
initial guess is x1 = 2, find the second approximation x2 .
[3]
(b) On the system of coordinates below, sketch the tangent line, a possible curve
y = f (x) and the two approximations x1 and x2 of the root.
[5]
12. Given the function f (x) = x, find the number c that satisfies the conclusion of the
√
Mean Value Theorem in the interval [0, 4]. On the diagram below, graph the function f ,
the secant line through the endpoints of the interval, and the tangent line at (c, f (c)).
4
3
2
1
0
−2 −1 0
−1
1
2
3
4
5
6
Math 151 Final Exam, Sample
9
[5]
13. Find the slope of the tangent line to the polar curve r = cos(2θ) when θ = π/4.
[7]
14. Find the point (x, y) of local minimum of the curve
x = t3 − 3t, y = t2 + t + 1
by using the second derivative test to verify that it is a minimum.