Differential Calculus Flashcards by Kent Jay Pepito

Set 1 Problem 43: Find the closest distance from the point (1,4) to the parabola y²=2x.

(2,2) p.13

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Set 1 Problem 47: A searchlight revolving once each minute is located at a distance of 1/4 mi from a straight beach. How fast (in mi/min), is the light moving along the beach when the beam makes an angle of 60 degrees with the shoreline.

2π (p.14)

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Set 1 Problem 63: The parabola y=x²+C has the line y=2x+4 as its tangent. Find the value of C.

5 (p.16)

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Set 1 Problem 65: Find the subnormal to the curve y=e^x at x=1.

e² (p.16)

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Set 2 Problem 9: Find the curvature of the curve given in equations x=sinΦ, y=sin2Φ at the point where Φ=(1/2)π.

1/4 (p.24)

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Set 2 Problem 27: The altitude of a certain right circular cone is the same as the radius of the base, and is measured as 5 inches with a possible error of 0.02 inch. Find approximately the percentage error in the calculated value of the volume.

1.2% (p.27)

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Set 2 Problem 36: The base of an isosceles triangle is 8 ft long. If the altitude is 6 ft long and is increasing 3 inches per minute, at what rate (in rad/min) are the base angles changing?

1/52 rad/min (p.29)

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Set 2 Problem 60: The height (in feet) at any time t (in seconds) of a projectile thrown vertically is: h(t)=-16t²+256t. Find its initial velocity.

256 ft/sec (p.32)

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Set 2 Problem 74: In the curve y=3cos(1/2)x, what is the amplitude and period?

3, 4pi (p.36)

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Set 3 Problem 2: Evaluate the limit of (Equation 1.1) as x approaches infinity.

1/4 (p.41)

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Set 3 Problem 4: Find the length of one arc of the curve represented by the parametric equation x = 4( θ - sin⁡θ ), y = 4( 1 - cos⁡θ ).

32 (p.42)

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Set 3 Problem 10: An object oscillates with Simple Harmonic Motion (SHM) according to the equation x=3sin(pi(t)) meters. Determine the amplitude, frequency, and period.

3m, 2 sec, 0.5 Hz (p.42)

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Set 3 Problem 28: The tangent to y=x^3-6x^2+8x at (3, -3) intersects the curve at another point. Find this point.

(0, 0) (p.46)

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Set 3 Problem 35: Solve the differential: xydx-(x+2y)^2dy=0; y(0)=2.

y^3(x+y)=16e^(x/y) (p.47)

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Set 3 Problem 37: An isosceles trapezoid has constant bases of 6 and 12 inches, respectively. Using differentials, find the approximate change in its area when the equal sides changes from 5 to 5.2 inches.

2.25 (p.47)

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Set 3 Problem 51: Find the slope of the curve r=3/(2 - cos⁡θ) at ⁡θ=π/2.

0.50 (p.50)

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Set 3 Problem 56: Find the asymptote of the curve y=(x³+1)/(x²-1).

y=x (p.52)

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Set 3 Problem 57: A kite at a height of 60 ft, is moving horizontally at a rate of 4 ft per second away from a boy who flies it. Find the rate of change of the angle of elevation of the kite when 100 ft of string are out.

-0.024 rad/s (p.52)

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Set 3 Problem 73: What is the period of the graph y = tan 2x?

π/2 (p.54)

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Set 3 Problem 74: Find the level curve of f(x,y)=x^2+4y^2 passing through (2,3).

6i+16j (p.54)

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Set 4 Problem 3: Transform the parametric equations x=costheta, y=cos^2theta+8costheta into its corresponding Cartesian equation.

x^2+8x-y=0 (p.60)

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Set 4 Problem 32: Find the angle between the curves xy = 4 and x^2 = 2y.

71.57° (p.64)

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Set 4 Problem 37: What is the derivative of 4cos(2-x^3)?

12x^2sin(2-x^3) (p.65)

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Set 4 Problem 65: A ship leaves a port at noon and travels due west at 20 mph. At 6:00 P.M., a second ship leaves the same port and travels northwest at 15 mph. How fast are the two ships separating when the second ship has traveled 90 miles?

12.44 mph (p.69)

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12.44 mph (p.69)

Set 4 Problem 66: A light is placed on the ground 60 ft from a building. A man 6 ft tall walks from the light toward the building at the rate of 5 ft/s. Find the rate at which his shadow on the wall of the building is changing when he is 30 ft from the building.

2 ft/s (p.70)

Set 4 Problem 67: Find the first derivative of y=x^lnx.

1-lnx^2x (p.70)

Set 4 Problem 68: Find the equation of the normal to the curve y+(x+y)^(1/2) = x at (3,1).

5x+3y=18 (p.70)

Set 5 Problem 1: Find the gradient of F(x, y, z) =4x²+x²y-2xy²-z³ at the point (3, -2, 1).

4i+33j-3k (p.77)

Set 5 Problem 10: What is the longest 1.8 m-wide shuffle board court that will fit in a 6 m x 9 m rectangular room?

9.017 m (p.78)

Set 5 Problem 14: A man on a wharf 25ft above the water pulls in a rope to which a raft is attached at the rate of 3 ft/s. At what rate is the raft approaching the wharf when there are 30ft of rope out?

-5.43 ft/s (p.80)

Set 5 Problem 29: The sum of two numbers is 18. Find the smaller of the 2 numbers if the product of one by the square of the other is a maximum.

6 (p.82)

Set 5 Problem 30: At noon a car drives from a town X towards the east of 60 mph. Another car starts from town Y towards town X at 30 mph. Town Y is 45 mi away and is N60degreesE from town X. Find the time after noon when the cars will be nearest each other.

29 min after noon (p.82)

Set 5 Problem 56: Evaluate the first derivative, y' when x=1 of Equation 5.5.

664 (p.85)

Set 5 Problem 57: Find the second derivative, d^2y/d⁡θ, when ⁡θ=0 if y=4sec2⁡θ.

16 (p.85)

Set 6 Problem 1: A box open at the top is to be made from tin 48 square ft. Find the maximum volume of the box that can be made.

32 ft^3 (p.94)

Set 6 Problem 3: Find the total differential of the given function z=x^2cos2y.

2xcos2ydx-2x^2sin2ydy (p.94)

Set 6 Problem 56: The base area and the height of a rectangular box are increasing at the rates 2 cm^2/s and 0.5 cm/s respectively. Find the rate at which volume of the box is increasing when the base area is 24 cm^2 and the height is 12 cm.

36 cm^3/s (p.103)

Set 6 Problem 57: Find the area of the largest rectangle that can be constructed with its base on the x-axis and two vertices on the curve Equation 6.3.

4a^2 (p.103)

Set 6 Problem 59: A spherical snowball is melting in such a way that its surface area decreases at the rate of 1 sq cm/min. How fast is its radius shrinking when it is 3 cm?

-1/(24pi) (p.103)

Set 6 Problem 64: What is the maximum area of a triangle inscribed in a semicircle with radius 12 cm if the hypotenuse of the triangle is on the diameter of the circle?

108.54 cm² (p.104)

Set 7 Problem 9: A lighthouse located on a small island 3 km away from the nearest point P on a straight shoreline and its line makes 2 revolutions per minute. How fast is the beam of the light moving along the shoreline when it is 1.2 km from P?

43.73 km/min (p.112)

Set 7 Problem 10: The volume of a sphere is increasing at a rate of 2 cm^2/s. How fast is the surface area of the sphere increasing when the radius is 8 cm?

0.5 cm^2/s (p.112)

Set 7 Problem 14: What is the 51st derivative of cosx?

sinx (p.113)

Set 7 Problem 16: Which of the following is the second derivative of x²-2y²=6?

-4/y² (p.113)

A particle is moving according to the law of the motion s=t^3-12t^2+36t, t≥0, where s is in meters and t is in seconds. Set 7 Problem 17: Find the velocity after 3 seconds.

-9 m/s (p.113)

A particle is moving according to the law of the motion s=t^3-12t^2+36t, t≥0, where s is in meters and t is in seconds. Set 7 Problem 18: For which time interval when the particle moving is speeding up?

26 (p.114)

A particle is moving according to the law of the motion s=t^3-12t^2+36t, t≥0, where s is in meters and t is in seconds. Set 7 Problem 19: Find the total distance traveled during the first 8 seconds.

96 m (p.114)

Set 7 Problem 29: Find the value of x for which the curve y=ln(x)/x^2 has a maximum point.

1.65 (p.115)

Set 7 Problem 30: Find the equation of the tangent of the line of the curve 2(x^2+y^2)^2=25xy^2 at the point (2,1).

11x-12y=10 (p.115)

Set 7 Problem 31: In a precision manufacturing process, ball bearings must be made with a radius of 0.60 mm with a maximum error in the radius of ±0.15 mm. Estimate the maximum error in the volume of the ball bearing.

±0.70 mm^3 (p.116)

Set 7 Problem 42: Find the equation of a curve if y"=3 and if it passes through (2,6) and (6,0).

y=1.5x^2+13.5x-27 (p.117)

Set 7 Problem 58: What is the radius of a closed cylindrical can with a volume of 169.646 in^3 that will require the minimum amount of material in making it?

3 in (p.120)

Set 7 Problem 62: Given the function Equation 7.2.Evaluate the limit of f(x) as x approaches to 3.

-1 (p.121)

Set 7 Problem 66: Find the curvature of the curve x²=8y at the point (4,2).

0.0884 (p.121)

Set 7 Problem 74: A window is to be formed by a rectangle surmounted by a semicircle. The perimeter of the window is to be 8.57 m. Find the width of the window that will the most amount of light.

2.4 m (p.123)

Set 8 Problem 5: Find the point of inflection of the curve y=x^2-x^-1.

(-1,0) (p.128)

Set 8 Problem 6: Find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 in by 24 in, and then turning up the sides.

486 in^3 (p.128)

Set 8 Problem 7: Find the volume of the largest right circular cylinder which can be inscribed in a sphere of radius 12 cm.

9.24 in (p.129)

Set 8 Problem 24: Find the maximum point of the curve (xsquared)y=(xcubed)-4.

(-2.-3) (p.132)

Set 8 Problem 25: Find the radius of curvature for the curve y=e^x at the point (0,1).

2√(2) (p.132)

Set 8 Problem 40: Represent the family of circles with their centers on the x-axis by a differential equation.

yy"+(y')²+1=0 (p.134)

Given the equation of the ellipse x^2+4y^2=16. Set 8 Problem 44: Find the second eccentricity of the ellipse.

1.414 (p.135)

Given the equation of the ellipse x^2+4y^2=16. Set 8 Problem 45: Find the center of curvature at the point (0,2).

(0,-6) (p.135)

Set 8 Problem 71: Given the equation y=ax^2+bx^2. Find the value of a if the curve has its point of inflection at (-1,2).

1 (p.139)

A particle moves along the path whose parametric equations are x=5t^2, y=6t-2. Set 9 Problem 45: Determine the velocity of the particle when t=5 seconds.

50.36 (p.152)

A particle moves along the path whose parametric equations are x=5t^2, y=6t-2. Set 9 Problem 46: Determine the acceleration of the particle when t=5 seconds.

10 (p.152)

A particle moves along the path whose parametric equations are x=5t^2, y=6t-2. Set 9 Problem 47: Find the distance travelled by the particle after t=5 seconds.

131 (p.152)

Set 9 Problem 69: If z=3x^2-2xy+7y^2, find the partial derivative of z with respect to x.

6x-2y (p.156)

Set 9 Problem 72: Differentiate y=lncos3x.

3tan3x (p.156)

Set 9 Problem 73: Find the slope of the curve whose polar equation is r=1+sintheta when theta=pi/3.

-1 (p.156)

Set 10 Problem 1: If A=3x^2-x , find dA for x=3 and dx=0.01.

0.17 (p.161)

Set 10 Problem 7: Find the area of the smallest ellipse circumscribed about a rectangle of sides 6cm by 4cm.

12pi sq. units (p.162)

Set 10 Problem 14: Find the angle between the tangents of the curves x²+y²=20 and y²=8x at their intersection.

71.57° (p.164)

Set 10 Problem 16: A hemispherical bowl is filled with water at a rate of 30 cm^3/s. The water surface is rising at a rate of 0.0159 cm/s when the depth of water in the bowl is 20 cm. Find the radius of the bowl.

25 cm (p.164)

Set 10 Problem 20: If y=x^5, what is the value of y^(4) when x=1.5?

180 (p.165)

Set 10 Problem 21: If y=x^5, what is the value of y^(4) when x=1.5?

5(2x+3)^(3/2) (p.165)

Set 10 Problem 25: Find the area of the triangle formed by the x-axis, the tangent, and the normal to the parabola x^2=8y at the point (4,2).

4 sq. units (p.166)

Set 10 Problem 28: Evaluate the limit of Equation 10.2 as x approaches pi.

-2 (p.166)

Set 10 Problem 33: If y=xe^x, find the value of d²y/dx² when x=2.

29.56 (p.167)

Set 10 Problem 35: A balloon is at a horizontal distance of 75m from an observer on a level ground. At a certain instant, the balloon is released and rises vertically at a rate of 3m/s. How rapidly will it be receding from the observer 20s later?

1.874m/s (p.168)

Set 10 Problem 36: A particle traveling a straight line has a velocity given by the law v(t)=6-2t+3t^2 and the distance (s) is equal to zero. How far has it travel after 5 seconds?

130 (p.168)

Set 10 Problem 49: The hour hand of the clock is 1 ft long and the minute hand is 1.6 ft long. At what rate in ft/min are the ends of the hands approaching each other at 2 o'clock pm?

0.095 in/min (p.170)

Set 10 Problem 64: A cylindrical tin tank, open at the top, has a copper bottom. The radius of the base is "r" and the height is "h". If sheet copper is 4 times as expensive as tin, per unit area, find the most economical proportions.

h=8r (p.172)

Set 10 Problem 69: Evaluate Equation 10.3.

0 (p.174)

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