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Name/Section
CALCULUS 1
December 13, 2007 FINAL EXAM
Directions. Please work as many problems as you can on the enclosed pages. The readers will read all of the problems and assign a numerical grade to each problem. Among the last seven problems the two with the lowest score will be deleted when the scores are added. (Problem #1 will not be deleted.) A maximum of 100 points can be earned on the exam.
It is important to show your work on all problems except #1 and #3. Correct answers on the other problems may not receive full credit if the reasoning and computational paths to the answers are not clearly indicated. (The readers of the exams will not assume responsibility for finding the next steps in haphazard presentations.) Please enclose each of your answers in a box.
Please work without the aid of notes, books, computers, calculators, and other people.
The course instructor may be in the room at the time of the examination acting as a resource. The examination is being given on an honor system.
1 2 3 4 5 6 7 8
Total
- (20 pts.) This problem consists of skills questions. Each part can earn up to two points. The two parts with the lowest score will be deleted. Please enter your answers in the boxes provided. No partial credit will be assigned for any of the problems. This problem contributes up to 20 points on this exam; additionally, its (normalized) score can be used to replace the skills quizzes average used to determine a course grade.
(a) Let f (t) = − 3 t^2 e−t, −∞ < t < ∞. Express in simplified form the value of f ′(1).
(b) Let f (x) = x^2 sec (x), −π 2 < x < π 2. Express in simplified form the value of f ′(π 4 ).
(c) Let h(t) = 8 sin ( t 3 t), 0 < t < ∞. Express in simplified form the value of h′(π).
(d) Let g(x) = 3 sin (x^2 ), −∞ < x < ∞. Express in simplified form the value of g′(
√π 2 ).
(j) Express in simplified form the value of
0 2 xe
−x^2 dx.
(k) Express in simplified form the value of
∫ (^) e 1
(ln x)^3 x dx.
(l) Express in simplified form the value of limx→∞ lnx^ x.
- (16 pts.) This problem consists of three parts.
(A.) (10 pts.) What is the slope of the tangent line to the curve modeled by y = ln x, 0 < x < ∞ at x = 1? Please box your answer.
(B.) (4 pts.) At what point on the natural exponential function is the tangent line parallel to the tangent line to the function ln x, 0 < x < ∞, at (2, ln 2)? Your boxed answer should be in the form of an ordered pair of real numbers.
(C.) (2 pts.) Complete the following definition by completing the equation. Definition. Let I denote an interval in R and let f denote a real- valued function defined on I. Let x 0 ∈ I. The derivative of f at the point, x 0 , denoted by f ′(x 0 ), is defined by
f ′(x 0 ) = lim ∆x→ 0
(7) A parametric curve is modeled by the vector-valued function r(t) = (x(t), y(t)), 0 ≤ t ≤ 1. The same parametric curve can also be modeled by a real-valued function, f (x), 0 ≤ x ≤ 1.
(8) Let f and g be two real-valued differentiable functions each defined on the interval [0, 1]. If f ′(x) = g′(x), 0 ≤ x ≤ 1 then f (x) = g(x), 0 ≤ x ≤ 1.
(8) Let f and g be two real-valued differentiable functions each defined on R. Then
d dx
f (g(x)) = f (g(x))g′(x), x ∈ R.
- (16 pts.) This problem consists of two parts, one of which consists of two parts. (A.) (12 pts.) Let r(t) = (t^3 , t), 0 ≤ t ≤ 4. The motion of a moving particle is modeled by r(t), where t denotes time. (i) Within the model find its speed at t = 2. Recall that speed and ve- locity vector are not the same, in this context. Please box your answer. (ii) Within the model find the position of the particle where its speed is a minimum. Your (boxed) answer should be in the form of a vector (element in R^2 ). (This problem can be solved by inspection.)
(B.) (4 pts.) The movement of a particle in the plane is modeled by r(t), 0 ≤ t ≤ 2 π, where t denotes time. The velocity v(t) of the parti- cle is modeled by v(t) = (cos (t), − sin (t)), 0 ≤ t ≤ 2 π. At time t = 0 the particle’s position is given by r(0) = (0, −1). What is the position of the particle at t = π 2? That is, compute r( π 2 ). Your (boxed) answer should be in the form of a vector (element in R^2 ).
- (16 pts.) On the xy-coordinate system provided, sketch a graph of the function, g, defined by
g(x) =
x (1 + x)^2
, −∞ < x < ∞, x 6 = − 1.
And, fill in the following table, leaving empty those blanks that don’t apply.
(list xy-coordinates) local maxima
(list xy-coordinates) local minima
(list xy-coordinates) inflections
(equations of) horizontal asymptotes
(equations of) vertical asymptotes
(a real number or ±∞) limx→∞ g(x)
(a real number or ±∞) limx→−∞ g(x)
y
x
- (16 pts.) A maximum of 16 points can be earned on this problem.
(A.) (6 pts.) Let y(x) = sin (x), 0 ≤ x ≤ π 4. Rotate about the x-axis the region of the plane bounded by the function y, the x- axis, the y-axis and the line x = π 4 , to obtain a solid. Model the volume of the solid in the form
V =
2 0
g(y) dy.
Find an explicit form for the function, g(y), 0 ≤ y ≤ √^12.
(B.) (6 pts.) The parametric curve modeled by the vector-valued function r(t) = (t^2 , t^4 + t^2 ), −∞ < t < ∞ can also be modeled by a scalar-valued function of the form f (x), 0 ≤ x < ∞. Find an explicit form for f (x).
(C.) (6 pts.) Let f (x) = sin (x^2 ), 0 ≤ x ≤ π 2. An accepted model for the arc length, `, of f can be written in the form
` =
∫ (^) b
a
g(x) dx.
Complete the boxed equation by finding values of a and b and an explicit form for g:
` =
dx.
