Download Winter 2008 Final Exam: Calculus Problems and more Exams Mathematics in PDF only on Docsity!
Question 1: (12 pts) For each of the following functions, find the derivative dy dx
You do not have to simplify your answers.
a) y = ln
x^4 cos^6 (3x + 2) √ (^3) x (^2) + 1
b) y = sec^3 (e^4 x−^1 ) c) 3 cot(x + y) = sin(y^2 ) d) y = e(^ x^12 ) cos−^1 (
x)
Question 2: (8 pts) For each of the following functions, find the second derivative d^2 y dx^2
. Simplify your answers as much as possible.
a) y = x^2 tan
x
b) y = x sin(ln(x))
Question 3: (4 pts) Find the equation of the line that is tangent to the graph of x^2 + y^2 = 25, at the point (3,-4).
Question 4: (4 pts) Find the slope of the line that is normal to the graph of
y =
(x^2 + 1)^2
at x = 1.
Question 5: (5 pts) Solve the equation cos(2x) − 3 x = −1, using Newton’s method. Give an answer that is accurate to four decimals, and start with a guess of x 1 = 1.
Question 6: (4 pts) The impedance Z (in Ω) in an electric circuit is given by Z =
R^2 + (XL − XC )^2. If R = 2500 Ω and XL = 1500 Ω, then find the value of XC that makes the impedance Z a minimum.
Question 7: (4 pts) The electrical potential on the line 3y − x = 2 is given by the function V = 4x^2 − 18 y^2 + 2. At what point of the line is the potential minimum?
Question 8: (30 pts) Evaluate the following integrals.
a)
x^4 − x^4 + ex^ −
e^4 dx b)
sin−^1 (x) dx c)
x^2 ex 3 dx
d)
3 x^2 ln(x) dx e)
x^2 sin(2x) dx f )
4 − 9 x^2
dx
g)
tan(ln(x)) x
dx h)
cos(x) 3 + sin^2 (x)
dx i)
1
2 x − 1
dx
j)
x + 2 x^2 + 4x + 5
dx
Question 9: (4 pts) Find the area enclosed by the curves y = x^2 and y = 2 − x.
Question 10: (5 pts) Give an estimate of
0
x^3 + 1 dx to four decimals, using
n = 4 and
a) the Trapezoidal Rule b) Simpson’s Rule
Question 11: (4 pts) In coming to a stop, the acceleration of a car is given by a(t) = − 4 t. The car is traveling at 32 m/s when it starts braking.
a) How long does it take for the car to stop?
b) What is the car’s braking distance?
Question 12: (4 pts) Find a 0 and b 3 of the Fourier series for the function
f (x) =
0 if −π 6 x < 0 x if 0 6 x < π
Question 13: (2 pts) Determine if the function y = x^4 + x + C ln(x) is a solution of the differential equation xy′′^ + y′^ = 16x^3.
Question 14: (4 pts) Find the solution of the differential equation y′^ = (1−y) cos(x), with the condition that y = 0 when x = π 6
Question 15: (6 pts) Find a general solution of the following differential equations.
a) y′^ = sin(x) sec(y)
b) y′^ − y = 3x
Answers
- a) y′^ =
x − 18 tan(3x + 2) − 2 x 3(x^2 + 1) b) y′^ = 12 sec^3 (e^4 x−^1 ) tan(e^4 x−^1 )
c) y′^ = −3 csc^2 (x + y) 3 csc^2 (x + y) + 2y cos(y^2 ) d) y′^ = −e^1 /x^2
1 − x
x
2 cos−^1 (
x) x^3
- a) y′^ = 2x tan
x
− sec^2
x
b) y′^ = sin(ln(x)) + cos(ln(x))
- y =
x −
