Download Final Exam Problems for Calculus 1 | MATH 161 and more Exams Calculus in PDF only on Docsity!
Millersville University Name Department of Mathematics MATH 161, Calculus I, Final Examination May 1, 2009, 08:00-10:00AM
Please answer the following questions. Your answers will be evaluated on their correctness, completeness, and use of mathematical concepts we have covered. Please show all work and write out your work neatly. Answers without supporting work will receive no credit. The point values of the problems are listed in parentheses.
- (4 points) Find the values of x for which the following function is discontinuous and clearly describe the types of discontinuities.
f (x) =
{ (^) x (^3) − 3 x (^2) + x^2 − 1 if^ x^6 =^ ±1, 9 if x = 1.
- (6 points) A rectangular box with no top is to be built by taking a 20-inch by 24-inch sheet of cardboard and cutting x-inch squares out of each corner and folding up the sides. Find the value of x which maximizes the volume of the box.
- (4 points each) Find the following derivatives. You do not need to simplify your results.
(a)
d dx
[cos(1 + 2 ln x)]
(b)
d dx
[ √ 4
x^2 − 1 + 2x^5
] 3
(c)
d dx
[ x
√ 3 − e1+cot^ x
]
(b) If the initial amount of 235 U present at the site of a test explosion of a nuclear weapon is denoted U(0), write an expression for the amount of 235 U remaining after t years.
(c) What fraction of one gram of 235 U will remain after 5 × 108 years?
- (4 points each) Find the exact values of the following limits if they exist. You must compute the limit algebraically, as opposed to plugging in numbers or using a graph to guess the answer. If a limit does not exist, please explain why.
(a) lim x→ 5
x^2 − 1 x^2 − 6 x + 5
(b) lim x→ 0 +^
x^3 ln(2x)
(c) Find the intervals on which f increases and the intervals on which f decreases.
(d) Find the x-coordinates of any local maxima or minima. If there are none, write “NONE”.
(e) Find the intervals on which f is concave up and the intervals on which f is concave down.
(f) Find the x-coordinates of any inflection points. If there are none, write “NONE”.
(g) Locate any horizontal or vertical asymptotes, and compute the relevant limits. If there are none, write “NONE”.
(h) Sketch the graph of the function on the grid provided below.
x
y
(d)
∫ (^4) π 2
π^2
cos
x √ x
dx
(e)
∫ ex
1 + ex^ dx
- (4 points) Show that the equation 1 = x^4 + 6x^2 has a solution lying in the interval [− 1 , 0].
- (4 points) Approximate the following definite integral using a Riemann sum with n = 40 subdivisions and right endpoint evaluation points. ∫ (^4)
1
sin(x^3 ) dx
