Functions
Condense into a single logarithm
f(x) = 5 ln(x-2) - ln(x+2) - 3 ln(x)
f(x) = ln((x-2)^5/x^3(x+2))
Solve for the point of intersection:
x + y + z = 3
x - 2y + 4z = 5
3y + 4z = 5
(5/3, 1/3, 1)
What is the length of the arc on a circle with radius 20 inches intercepted by a central angle of 138 degrees?
Round to the nearest hundredth.
48.17 inches
Find the angle between
u=<-1, 3, 0>
v=<1, 2, -1>
in degrees rounded to nearest hundredth.
49.80 degrees
Evaluate lim x-> 3 of f(x) = x^3 - 2x + 1
22
f(2) = -5 and f(-1) = 4
f(x) = -3x + 1
Solve for x:
2x2 + 5x > 12
{x > 3/2} U {x < -4}
A periodic function starts at its minimum and completes one full cycle ever 10 units. The minimum value is -4 and the maximum value is 18. Find a formula for this function.
f(x) = -11 cos((pi/5)x) + 7
Find the standard form of the complex number
6[cos(18o 45') + i sin(18o 45')]
Round to 4 decimal places.
5.6816 + 1.9286i
Find the derivative of
f(x) = x2 + 3x - 7
f'(x) = 2x + 3
f(x) = 2(x-4)2
Restrict the domain so that the inverse of f is a function, and find the inverse function.
Domain: {x | x>=4}
f-1(x) = sqrt(x/2) + 4
Solve for the exact value of x:
e2x - 6ex + 8 = 0
x = ln(4)
x = ln(2)
Find the linear and angular speed of a record which is 16 inches in diameter and plays at 34 revolutions per minute.
Answer in inches per second and radians per second, rounded to the nearest hundredth.
Angular speed = 3.56 radians/s
Linear speed = 28.48 inches/s
Find the projection of u onto v, then write u as the sum of orthogonal vectors, one of which is the projection of u onto v.
u=<-3, -2>
v=<-4, -1>
u = <-56/17, -14/17> + <5/17, -20/17>
Find the slope of 3x2 - 5x + 2 at x = -1
-11
Find the domain, zeros, vertical and horizontal asymptotes of
f(x)= (x^2 - x - 2) / (x^3 - 2x^2 - 5x + 6)
domain: {x | x not = 1, 3, -2}
zeros: x = {2, -1}
vert. asymp.: x=1, x=3, x=-2
horiz. asymp.: y = 0
Solve for x (rounded to nearest thousandth):
log6(x+2) - log6(x) = log6(x+5)
x = 0.449
or
x = sqrt(6) - 2
Verify the identity:
(tan(x) + cot(y)) / (tan(x)cot(y)) = tan(y) + cot(x)
Start with left side. Split fraction into a sum. Divide out common factors. Convert to reciprocal functions.
Find all 5 fifth roots of 1.
Round to 4 decimal places.
1
0.3090 +- 0.9511i
-0.8090 +- 0.5878i
Find the equation of a line tangent to
f(x) = -0.6x2 + 0.3x + 2.1
at x = 0.2
y - 2.136 = 0.06(x - 0.2)
Find all zeros of f(x) given that (2+i) is a zero.
f(x) = 2x4 - 3x3 - 13x2 + 37x - 15
x = 2 + i
x = 2 - i
x = -3
x = 1/2
Solve for x:
3x/(x-1) <= x/(x+4) + 3
{x < -4} U {-2<=x<1} U {x>=6}
Solve for all values of x (in radians):
cos(x) + sin(x)tan(x) = 2
x = pi/3 + 2kpi
x = 5pi/3 + 2kpi
Find a unit vector orthogonal to both u and v.
u= -3i + 2j - 5k
v= 10i - 15j + 2k
1/sqrt(7602) <-71, -44, 25>
Find the slope-intercept form of the equation of a line tangent to f(x) = sqrt(3x - 1) + 2 when x = 4.
Round to 4 decimal places.
y = 0.4522x + 3.5074