Differential Calculus

Find the slope of y = 1 – x3 at the point where y = 9. a. –11 b. –10 c. –12 * d. –13 Find the derivative of y with respect to x if y = xlnx – x. a. 1 b. lnx * c. x d. lnx – 1 For what value of x will the curve y = x3 – 3x2 + 4 be concave upward? a. 1 b. 3 c. 2 * d. 4 How fast does the diagonal of a cube increases if each edge of the cube increases at a constant rate of 5 cm/s? a. 6.7 cm/s b. 7.7 cm/s c. 8.7 cm/s * d. 9.7 cm/s If f(x) = tanx – x and g(x) = x3, evaluate the limit of f(x)/g(x) as x approaches zero. a. 0 b. ∞ c. 3 d. 1/3 * Find the derivative of y = xlnx. a. –1/x b. –1/x2 * c. –1/x3 d. –1 If xy3 + x3y = 2, find dy/dx at the point (1,1). a. 1 b. –1 * c. 2 d. –2 The tangent line to the curve y = x3 at the point (1,1) will intersect the x-axis at x = a. 2/3 * b. 4/3 c. 1/3 d. 5/3 If y = ex + xe + xx, find y’ at x = 1. a. e + 1 b. e – 1 c. 2e + 1 * d. 2e – 1 Find the value for which y = x3 – 3x2 has a minimum value. a. 1 b. 2 * c. 0 d. –2

c. tan-1(2/3)

d. tan-1(3/4) *

c. 2 cm2/min *

d. 5 cm2/min

If z = xy2, and x changes from 1 to 1.01, and y changes from 2 to 1.98, find the approximate change in z. a. –0.0202 b. –0.0303 c. –0.0404 * d. –0.0505

Find the equation of the line tangent to y = x2 – 3x – 5 and parallel to the line y = 3x – 2. a. y = 3x – 14 * b. y = 3x – 13 c. y = 3x – 12 d. y = 3x – 11

If y = ln(tanhx), find dy/dx. a. 2sech2x b. 2sech2x c. 2csch2x * d. 2coth2x

A rectangular field is fenced off, an existing wall being used as one side. If the area of the field is 7,200 sq. ft, find the least amount of fencing needed. a. 250 ft b. 240 ft * c. 230 ft d. 220 ft

Find the approximate surface area of a sphere of radius 5.02 cm. a. 317 sq. cm * b. 315 sq. cm c. 313 sq. cm d. 311 sq. cm Find the value of x for which y = x5 – 5x3 – 20x – 2 will have a maximum point. a. –1 b. –2 * c. 1 d. 2 If y = ln(x2ex), find y”. a. –1/x2 c. –1/x

b. –2/x2 * d. –2/x

Find the radius of curvature of y = x3 at the point (1,1). a. 3.25 b. 4.26 c. 5.27 * d. 6.28

The side of an equilateral triangle is increasing at the rate of 0.50 cm/s. Find the rate at which its altitude is increasing. a. 0.334 cm/s b. 0.443 cm/s c. 0.433 cm/s * d. 0.343 cm/s At what acute angle does the curve y = 1 – ½x2 cut the x-axis? a. 34.54° b. 44.64° c. 54.74° * d. 64.84° Find the equation of the line with slope –1/2 and tangent to the ellipse x2 + 4y2 = 8. a. x + 2y – 4 = 0 * b. x – 2y + 4 = 0 c. x + 2y + 4 = 0 d. x – 2y – 4 = 0

Find the second derivative (y”) of 4x2 + 9y2 = 36 by implicit Find the point on the curve y = x3 – 3x for which the tangent line is differentiation. parallel to the x-axis. a. –16y3/9 b. –16/9y3 * a. (-1,2) * b. (2,2) c. –9y3/16 d. –9/16y3 c. (1,2) d. (0,0) If f(x) = e x – e-x – 2x and g(x) = x – sinx, evaluate the limit of If y = ½tan2x + lncosx, find y’. f(x)/g(x) as x approaches zero. a. tan3x * b. tanx – sinx a. ∞ b. 0 c. tanx sec2x d. 0 c. 1 d. 2 * Find two numbers whose sum is 8 if the product of one number and the cube of the other is a maximum. a. 3 and 5 b. 4 and 4 c. 2 and 6 * d. 1 and 7

The volume of a cube is increasing at the rate of 5 cm3/min. How fast is the surface area increasing when the length of each edge is Find the angle of intersection between the curves y = x2 and x = y2. 12 cm? a. tan-1(1/4) b. tan-1(1/3) a. 3 cm2/min b. 4 cm2/min

The volume of a sphere is increasing at the rate of 6 cm3/hr. At what rate is its surface area increasing when the radius is 40 cm? a. 0.30 cm2/hr * b. 0.40 cm2/hr c. 0.50 cm2/hr d. 0.60 cm2/hr Find the point of inflection of y = 4 + 3x – x3. a. (1,6) b. (0,4) * c. (-2,4) d. (2,2)

Find the slope of the tangent to the curve xy2 + √xy = 2 at the point (1,1). a. –1/5 b. –2/5 c. –3/5 * d. –4/5

If x increases at the rate of 30 cm/s, at what rate is the expression (x + 1)2 increasing when x becomes 6 cm? a. 400 cm2/s b. 410 cm2/s c. 420 cm2/s * d. 430 cm2/s

If y = ½x[sin(lnx) – cos(lnx)], find dy/dx. a. sin(lnx) * b. cos(lnx) c. –sin(lnx) d. –cos(lnx)

What is the maximum value of y = 3 sinx + 4 cosx? a. 8 b. 7 c. 6 d. 5 *

If x = et and y = 2e-t, find the d2/dx2. a. 4e-t b. 4e-2t c. 4e-3t * d. 4e-4t

Find the maximum point of the curve y = 4 + 3x – x3. a. (-2,6) * b. (0,4) c. (1,6) d. (-3,22)

Two corridors 6m and 4m wide respectively, intersect at right angles. Find the length of the longest ladder that will go horizontally around the corner. a. 13 m b. 14 m * c. 15 m d. 16 m

Water flows into a cylinder tank at the rate of 20 m3/s. How fast is the water surface rising in the tank if the radius of the tank is 2 m? a. 5/pi * b. 6/pi c. 3/pi d. 4/pi

If y = 4/(2x – 1)3, find y” at x = 1. a. 190 b. 191 c. 192 * d. 193

If (0,4) and (1,6) are critical points of y = a + bx + cx3, find the value of c. a. 1 b. 2 c. –1 * d. –2

The side of an equilateral triangle increases at the rate of 2 cm/hr. At what rate is the area of the triangle changing at the instant when the side is 4 cm? a. 3√3 b. 4√3 * c. 5√3 d. 6√3

Let f be a function defined by f(x) = Ax2 + Bx + C with the following properties: f(0) = 2, f’(2) = 10 and f”(10) = 4. Find the value of B. a. 1 b. 2 * c. 3 d. 4

Find the value of x and y which satisfy 2x + 3y = 8 and whose product is a minimum. a. 1 and 2 b. 3 and 2/3 c. 3/2 and 5/3 d. 2 and 4/3 *

An open box is made by cutting squares of side x inches from four corners of a sheet of cardboard that is 24 inches by 32 inches and then folding up the sides. What should x be to maximize the volume of the box? a. 16.3 in b. 15.2 in c. 13.8 in d. 14.1 in *

If x = 2 sinθ, y = 1 – 4cosθ, then dy/dx is equal to a. 2 cotθ b. 2 tanθ * c. 2 cscθ d. 2 secθ

Two post 30 m apart are 10 m and 15 m high respectively. A transmission wire passing through the tops of the posts is used to brace the posts at a point on level ground between them. How far from the 10 m post must that point be located in order to use the least amount of wire? a. 10 m b. 11 m c. 12 m * d. 13 m The sum two numbers is K. Find the minimum value of the sum of their cubes. a. K3 b. K3/2 c. K3/3 d. K3/4 * Find the value of x so that the determinant given below will have a minimum value. 212 34x X31 a. 5 b. 6 c. 7 * d. 8 A closed right circular cylinder has a surface area of 100 cm2. What should be its radius in order to provide the largest possible volume? a. 3.320 cm b. 2.330 cm c. 3.203 cm d. 2.303 cm * Using differentials, determine the appropriate increase in the volume of a sphere if the radius increases from 5 cm to 5.05 cm. An ellipse has an equation of 9x2 +16y2 =144. Find the length of the shortest line segment in the first quadrant that can be drawn tangent to the ellipse and meeting the coordinate axes. An ellipse has an equation of 9x2 +16y2 =144. Compute the maximum area of a rectangle that could be inscribed in the ellipse?

Three sides of trapezoid are each 8 cm long. How long is the fourth Find the point on the curve y2=8x which is nearest to the external side when the area of the trapezoid has the largest value? point (4,2). The upper and lower edges of a picture frame hanging on a wall are a. 14 cm b. 15 cm 8 feet and 2 feet above an observer’s eye level respectively. How c. 16 cm * d. 17 cm In triangle MNO, MN=4.25 cm, NO=9.61 cm and OM=8.62 cm. A far from the wall must the observer stand in order that the angle rectangle is inscribed in it such that its shorter side is on the 4.25 subtended by the picture is a maximum? A sector with perimeter of 24 cm is to be cut fro a circle. What cm side of the triangle. Find the maximum area of the rectangle. a. 3.5 ft b. 4 ft * should be the radius of the circle if the area of the sector is to be a c. 4.5 ft d. 5 ft maximum? A wall 3 m high is 2.44m away from a building. What is the length a. 6 cm * b. 7 cm in m, of the shortest ladder that can reach the building with one end c. 5 cm d. 4 cm resting on the ground outside the wall?

A closed cylindrical tank was built with minimum surface area. Determine the ratio of its altitude to its radius. If h=65 tan Φ, what is the approximate change in h when Φ changes from 60° to 60°03’? She sum of two numbers is 10. Find the numbers such that their product is to be maximum. a. 5 𝑎𝑛𝑑 5 b. 4 𝑎𝑛𝑑 6 c. 10 𝑎𝑛𝑑 2 d. 2 𝑎𝑛𝑑 8

Given the curve 𝑦 2 = 5𝑥 − 1 at point (1, −2), find the equation of tangent and normal to the curve. a. 5𝑥 + 4𝑦 + 3 = 0 & 4𝑥 − 5𝑦 − 14 = 0 b. 5𝑥 + 4𝑦 − 3 = 0 & 4𝑥 + 5𝑦 − 14 = 0 c. 5𝑥 − 4𝑦 + 3 = 0 & 4𝑥 + 5𝑦 + 14 = 0 d. 5𝑥 − 4𝑦 − 3 = 0 & 4𝑥 + 5𝑦 − 14 = 0 The volume of a cube is increasing at the rate of 5 𝑐𝑚3 /𝑚𝑖𝑛. How fast is the surface area increasing when the length of each edge is 12 cm? a. 3 𝑐𝑚2 /𝑚𝑖𝑛 b. 4 𝑐𝑚2 /𝑚𝑖𝑛 2 c. 2 𝑐𝑚 /𝑚𝑖𝑛 * d. 5 𝑐𝑚2 /𝑚𝑖𝑛

Find y’ if 𝑦 = arcsinh cot 𝑥 a. csc 𝑥 c. csc 𝑥 cot 𝑥

Limit of a function: Evaluate: lim

𝑥 2 −16

c. (-2,4)

𝑥→4 𝑥−4

a. 0 c. 1

General Applications:

b. 8 * d. 16

Evaluate: lim

3

Find the second derivative of y = x(x + 1) , when x = 1 a. 42 b. 57 c. 36 * d. 16

Two posts one is 10 m high and the other 15 m high stand 30 m apart. They are to be stayed by transmission wires attached to a single stake at ground level, the wires running to the top of the posts. Where should the stake be placed to use the least amount of Tangents and Normal to Polynomial Curves: wire. b. 14 𝑚 The tangent line to the curve 𝑦 = 𝑥 3 at the point (1,1) will intersect a. 12 𝑚 * c. 18 𝑚 d. 16 𝑚 the x-axis at x = a. 2𝑥 − 𝑦 = −4 * b. −2𝑥 − 𝑦 = 4 The upper and lower edges of a picture frame hanging on a wall are c. 2𝑥 + 𝑦 = 4 d. −2𝑥 + 𝑦 = −4 8 feet and 2 feet above an observer’s eye level respectively. How Find the equation of the normal line to the graph of 𝑦 2 = 𝑥 2 + 3at far from the wall must the observer stand in order that the angle subtended by the picture is a maximum? the point (-1, 2) a. 3.5 ft b. 4 ft * a. 2/3 * b. 4/3 c. 4.5 ft d. 5 ft c. 1/3 d. 5/3

𝑥 3 −2𝑥+9

𝑥→∞ 2𝑥 3 −8

a. 0 c. 1/2 *

b. 2 d. 1/4

Evaluate: lim

𝑥→0

1−cos 𝑥 𝑥2

a. 0 c. 2

b. 1/2 * d. -1/2

Derivative of a function: 1

Differentiate: (𝑥 2 + 2)2 1

a. c.

(𝑥 2 +1)2 2 2𝑥 1 (𝑥+2)2

d. (2,2)

b. − csc 𝑥 * d. − csc 𝑥 cot 𝑥

b. d.

𝑥 1

*

(𝑥 2 +2)2 (𝑥 2

+ 2)2

Find the derivative of y with respect to x if y = xlnx – x. a. 1 b. lnx * c. x d. lnx – 1

Minima, Maxima and Point of Inflection: Given the curve 𝑦 = 12 − 12𝑥 + 𝑥 3 ,determine the maximum, minimum and inflection points. a. (−2 , 28 ), ( 2 , −4 ), & ( 0 , 12 ) b. ( 2 , −28 ), ( 2 , 4 ), & ( 0 , 2 ) c. (−2 , −28 ), (−2 , −4 ), & ( 2 , 12 ) d. (−2 , 28 ), (−2 , 4 ), & ( 1 , 12 )

Find a point on the curve 𝑥 2 = 2𝑦 which is nearest to a point (4, 1) a. (2, 4) b. (4, 2) c. (2, 2) * d. (2, 3) Find the slope of 𝑦 = 1 − 𝑥 3 at the point where y = 9. a. –11 b. –10 c. –12 * d. –13 Find the radius of curvature of 𝑦 = 2𝑥 3 − 𝑥 + 3 at the point (1,1). a. 10.048 b. 13.048 c. 11.048 * d. 15.048

If y = ln(tanhx), find dy/dx. a. 2sech2x b. 2sech2x c. 2csch2x * d. 2coth2x

Find the value for which 𝑦 = 𝑥 3 – 3𝑥 2 has a minimum value. a. 1 b. 2 * c. 0 d. –2

Differentiate: 𝑦 = 2 sin 𝑥 cos 𝑥 a. 2 sin 2𝑥 c. − sin 2𝑥

b. 2 cos 2𝑥 * d. − cos 2𝑥

Find y’ if 𝑦 = arcsin cos 𝑥 a. −1 * c. cos sin 𝑥

Find the value of x for which 𝑦 = 𝑥 5 − 5𝑥 3 − 20𝑥 − 2 will have a maximum point. a. –1 b. –2 * c. 1 d. 2

How fast does the diagonal of a cube increases if each edge of the cube increases at a constant rate of 5 cm/s? a. 6.7 cm/s b. 7.7 cm/s c. 8.7 cm/s * d. 9.7 cm/s

b. −2 d. − sin cos 𝑥

Find y’ if 𝑦 = cosh2 𝑥 − sinh2 𝑥 a. 2 cosh 2𝑥 c. 1

b. 2 sinh 2𝑥 d. 0 *

Find two numbers whose sum is 8 if the product of one number and the cube of the other is a maximum. a. 3 and 5 b. 4 and 4 c. 2 and 6 * d. 1 and 7

Water is flowing into a conical cistern at the rate of 8 𝑚3 /𝑚𝑖𝑛. If the height of the inverted cone is12 m and the radius of its circular opening in 6 m. How fast is the water level rising when the water is 4 m depth. a. 0.74 𝑚/𝑚𝑖𝑛 b. 0.64 𝑚/𝑚𝑖𝑛 * c. 0.54 𝑚/𝑚𝑖𝑛 d. 0.84 𝑚/𝑚𝑖𝑛

Find the point of inflection of 𝑦 = 4 + 3𝑥 − 𝑥 3 . a. (1,6) b. (0,4) *

Time Rates:

Water drains from a hemispherical basin of 20 inches at the rate of 3 𝑖𝑛3 /𝑠. How fast is the water level falling when the depth of water is 5 inches? a. 0.04693 in/s b. 0.01273 in/s * c. 0.09623 in/s d. 0.01732 in/s