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WorksheetsDiff Cal & Analytic Geom(#18)
Total questions: 55
Worksheet time: 28mins
If y = 3eˣ sin 2x, what is the value of x so that y’ = 0?
3pi/2
pi/2
pi/3
2pi/3
Find the second derivative of the function y = 5x cubed + 2x + 1.
2x
x
30x
24x
If y = arctan (ln x), find y’ when x = 1/e.
e/2
e/3
e
e/4
Find dy/dx, given y = u³ + 4 and u = x² + 2x.
6x²(x² + 2)²(x + 1)
3x²(x² + 2)²(x + 1)
3x²(x + 2)²(x + 1)
6x²(x + 2)²(x + 1)
If x³ – 3xy + y³ = 1, find y’.
(x² – y)/(y² – x)
(x² – y²)/(x² – y)
(x² – y)/(x + y)
(x² – y)/(x – y²)
Suppose f and g are functions such that f(2) = –1, f’(2) = 4, f’’(2) = –2, g(2) = –3, g’(2) = 2 and g’’(2) = 1. Find the value of (2f – 3g)’ at x = 2.
6
9
–7
–8
Suppose f and g are functions such that f(2) = –1, f’(2) = 4, f’’(2) = –2, g(2) = –3, g’(2) = 2 and g’’(2) = 1. Find the value of (fg)’ at x = 2.
–14
–12
–15
–13
Find the first partial derivative of u = xy + yz + zx.
y – z, x – z, y + x
y + z, x – z, x + y
y – z, x + z, y – z
y + z, x + z, y + x
Find the total differential of 3x³ + 4x²y – 2y³.
(9x² + 8xy) dx
(9x² + 8xy) dx + (4x² – 6y²) dy
(36x² + 8xy) dx + (3x² – 16y²) dy
(4x² – 6y²) dy
Find du/dt if u = x² – 2xy + y², x = (t + 1)², y = (t – 1)².
32t
2(t – 1)
64t
2(t + 1)
Evaluate lim (x² – 4) / (x – 2) as x → 2.
4
6
8
16
Evaluate lim x sin (pi/x) as x approaches to infinity.
1
pi/2
pi
infinity
Evaluate lim (z squared + 1)/(z raised to the 6 + 1) as z approaches to i.
sq. rt of 2(1 + i)/2
(–4/3) – 4i
1/3
–12 + 6i
Find the limit of (sin n)/n as n approaches infinity.
indeterminate
no limit
0
1
Evaluate lim (2 – x)^tan(pi x/2) as x → 1.
e^(2/pi)
e^(pi/2)
e^(2pi)
0
If y = 4 cosx + sin 2x, what is the slope of the curve when x = 2?
–2.21
–4.94
–3.25
2.21
Find the slope of r = 1/theta when theta = pi.
1/4
–pi
1/2
2pi
Find k so that the line 4x – y + 3 = 0 is tangent to the curve x sq. – y + k = 0.
4
11
0
7
Find the equation of the line tangent to the graph at the tangent point: (x + y) cubed = x + y + 6, (3, –1).
3x – y = 6
2x – y = 1
x + y = 3
x + y = 2
Locate and classify the critical points of y = x² – 4x – 1.
(2, –5) max
(2, –5) min
(5, –2) min
(5, –2) max
Find the value of x for which f(x) = x² + 5x + 2 is maximum.
5/2
2
–2
–5/2
Find the critical numbers of f(x) = x⁶/5 – 24x¹/5.
4
0
0, 4
0, 3
Find the point of inflection for the curve y = 3(x to the 4th power) – 8(x cubed) + 6(x squared).
(1/2, 11/24)
(1/4, 1/2)
(1/5, 1/8)
(1, 1)
Find the equation of the line normal to the curve y = 3x⁵ + 10x³ + 15x + 1 at its point of inflection.
x + 15y = 15
x – 15y = 15
15x – y = 15
x + 15y = –15
A rectangle with sides parallel to the coordinate axes has one vertex at the origin, one on the positive x-axis and its fourth vertex in the first quadrant on the line with equation 2x + y = 100. What is the maximum possible area of the rectangle?
3520
1250
1908
2250
A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum?
3.41 m
3.51 m
3.71 m
4.41 m
Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm.
11.55 cm
14.55 cm
12.55 cm
18.55 cm
A steel girder 8 m long is moved on rollers along a walkway 4 m wide and into a corridor perpendicular to the walkway. How wide must the corridor be to successfully move the girder?
1.8 m
18 m
10 m
8 m
The equation of motion of a car moving in a straight line is given by s = 18t + 19t² where s is the displacement in meter at any time t. Find the velocity of the car after 3 seconds.
123 m/s
132 m/s
168 m/s
186 m/s
A particle’s position (in inches) along the x-axis after t seconds of travel is given by the equation x = 24(t squared) – t cubed + 10. What is the particle’s average velocity during the first 3 seconds of travel?
72
48
32
63
A man is driving a car at the rate of 30 km/hour towards the foot of monument 6 m high. At what rate is he approaching the top when he is 36 m from the foot of the monument?
–52.80 km/hr
10.55 km/hr
–29.59 km/hr
12.52 km/hr
One end of a 32-meter ladder resting on a horizontal plane leans on a vertical wall. Assume the foot of the ladder to be pushed towards the wall at the rate of 2 meters per minute. How fast is the top of the ladder rising when its foot is 10 meters from the wall?
+0.568 m/min
+0.658 m/min
+0.896 m/min
0.986 m/min
Sand is being poured into a conical pile in such a way that the height is always 1/3 of the radius. At what rate is sand being added to the pile when it is 4 ft. high if the height is increasing at 2 in/min?
100,132.88 in³/min
53,288.13 in³/min
30,288.13 in³/min
130,288.13 in³/min
Find the approximate change in volume of a cube if you increase the side from 2 to 2.05 units.
0.4
0.5
0.6
0.3
The specific gravity of an object more dense than water is given by S = A/(A – W), where A is the weight in air and W is the weight in water. If, for a certain object, the measurements are A = 12 lbs and W = 5 lbs. The maximum error for air is ±1/2 ounce and ±1 ounce for water. What is approximately the maximum error in the computed value for S?
± 0.0175
± 0.0185
± 0.0195
± 0.0205
Find the radius of curvature of the parabola y² – 4x = 0 at the point (4, 4).
22.36
20.36
25.36
27.36
Find the center of the curvature for the curve y = (x to the power x) at P(0, 1).
(–2, –2)
(–2, 3)
(2, –3)
(2, 3)
it represents the distance of a point from the y-axis.
abscissa
ordinate
coordinate
polar distance
find the point on the line 3x + y + 4 = 0 that is equidistant from the points (–5, 6) and (3, 2).
(–2, 2)
(–2, 3)
(–2, –2)
(2, 2)
find the distance between the line x + y = 2 and a given point (1/2, 1/3).
7(sq. rt. of 2)/12
5(sq. rt. of 2)/6
12(sq. rt. of 2)/7
6(sq. rt. of 2)/5
find the distance between the lines, 3x + y – 12 = 0 and 3x + y – 4 = 0
16/sq. rt. of 10
12/sq. rt of 10
4/sq. rt. of 10
8/sq. rt. of 10
determine the point of division of the line segment from a(5, 6) to b(–3, –2) that divides this line segment, starting from a, into two parts in the ratio 1 : 3.
(1, 3)
(–1, 1)
(0, 2)
(3, 4)
find the new coordinates of the point p(3, 5) if the origin is moved to (2, 4) by a translation.
(1, 1)
(2, 2)
(1, 2)
(2, 1)
find the point to which origin must be translated in order that the transformed equation of xy – 2x – 4y – 4 = 0 shall have no first-degree term.
(2, 4)
(4, 4)
(4, 2)
(4, 3)
find the area of the region inside the triangle with vertices (1, 1), (3, 2) and (2, 4).
5/2
3/2
1/2
7/2
find the area of the polygon with vertices at 2 + 3i, 3 + i, –2 – 4i, –4 – i, –1 + 2i.
47
47/2
25
25/2
a line is perpendicular to the y-axis has a slope equal to:
one
infinity
zero
indeterminate
the product of the slopes of any two straight lines is negative 1, one of the lines is said to be
perpendicular
parallel
non intersecting
skew
the acute angle between the two straight lines y = 3x + 2 and y = 4x + 7 is close to
0 deg
90 deg
4.399 deg
28.30 deg
Determine the equation of the line passing through the points (1, 17) and (13, 4).
13x – 12y – 217 = 0
13x – 12y + 217 = 0
13x + 12y – 217 = 0
13x + 12y + 217 = 0
Express (3/4)x + (5/2)y = 3/2 in intercept form.
x/(3/5) + y/2 = 1
x/2 + y/(3/5) = 1
x/2 + (5/3)y = 1
x/2 + y/(5/3) = 1
Find the equation of the line with slope = 3 and the y-intercept = –2.
y = 3x – 2
y = –3x + 2
y = 3x + 2
y = –3x – 2
Find the equation of the line passing 3 units from the origin and parallel to 3x + 4y – 10 = 0.
3x + 4y – 5 = 0
4x + 3y + 1 = 0
x – 3y + 15 = 0
3x + 4y – 15 = 0
Find the equation of the line which is perpendicular to the line 3x – 2y – 5 = 0 and which passes through the point of intersection of 4x – y – 5 = 0 and x – y – 5 = 0.
2x – 3y + 2 = 0
2x – 3y – 15 = 0
2x + 3y + 15 = 0
2x + 3y – 2 = 0
The points A(1, 0), B(9, 2) and C(3, 6) are the vertices of a triangle, which of the following is an equation of one of the medians?
4x – 5y = 6
4x + 5y = 4
4x – 5y = 4
4x + 5y = 6
