Differential Calculus Flashcards
Determine the derivative of (x + 3)² + x³ with respect to x.
a) 3x² + 2x + 3
b) 3x² + 2x + 6
c) 3x² + 2x - 3
d) 3x² + 2x - 6
b) 3x² + 2x + 6
Determine the derivative of (x² - 1)(2x² - 3) with respect to x.
a) 4x(4x² - 5)
b) 2x(4x² - 5)
c) 2x(4x² + 5)
d) 4x(4x² + 5)
b) 2x(4x² - 5)
Determine the derivative of (x + 2)³ / x.
a) [3x(x + 2) + (x + 2)³] / x²
b) [3x(x + 2)² - (x + 2)³] / x²
c) [3x(x + 2)² + (x + 2)³] / x²
d) [3(x + 2)² - (x + 2)³] / x²
b) [3x(x + 2)² - (x + 2)³] / x²
Determine the derivative of sin⁴3x.
a) 12 sin³3x cos 3x
b) 4 sin³3x cos3x
c) 12 sin³3x
d) 4 sin³3x
a) 12 sin³3x cos3x
Given that y = x tan²x, find y’ when x = 2.
a) 3.66 ⨉ 10³
b) -45.69
c) -0.212
d) 5.61
b) -45.69
Determine the derivative of log(3x + 5).
3 / [(3x + 5)(ln 10)]
Determine the derivative of ln|2x² - 2x|.
(2x - 1) / (x² - x)
Determine the derivative of 2³ˣ.
2³ˣ ln 8
Determine the derivative of e⁵ˣ ⁺ ¹.
5e⁵ˣ ⁺ ¹
Find dy/dx of the function x² + xy + 2y² + 3x - 2y + 5 = 0.
dy/dx = (-2x - y - 3) / (x + 4y -2)
What is the slope of the curve x² + y² - 6x + 10y + 5 = 0 at (1, 0)?
a) 2.5
b) 5/2
c) -2/5
d) -5/2
a) 2/5
What is the equation of the tangent line to the curve x² + y² - 6x + 10y + 5 = 0 at (5, 0)?
2x + 5y = 10
Find the maximum value of y = 4sinx + 3cosx.
a) 5
b) 4
c) 4.95
d) 7
a) 5
A window is in the shape of a rectangle, surmounted by a semi-circle. If the perimeter of the window is 20 ft, what is its maximum area?
a) 24
b) 26
c) 28
d) 30
c) 28
A cylindrical can is to contain 2000 in³ of liquid. What height will minimize the cost of metal to be used in the construction of the can?
a) 8.60 in
b) 17.20 in
c) 6.83 in
d) 13.66 in
d) 13.66
A rectangular poster, which contain 50 in² of print, must have a margin of 2 in on each side and 4 in on top and bottom. What height will maximize the amount of material used?
a) 18 in
b) 16 in
c) 15 in
d) 12 in
a) 18 in
A triangle has a base of 24 m and an altitude of 18 m. A rectangle is inscribed in it such that its base coincides with the 24 m base. Determine the largest area of the rectangle.
a) 118 m²
b) 108 in²
c) 90 in²
d) 135 in²
b) 108 in²
From the same starting point, a snail and a turtle run at the same time in perpendicular directions for 3 kph and 4 kph, respectively. How fast is their distance changing after 2 hours?
a) 4.5 kph
b) 5 kph
c) 5.5 kph
d) 6 kph
b) 5 kph
A man, 1.6 m tall is walking on a horizontal street at 3 kph away from a vertical streetlight, 3 m high. How fast is the length of his shadow increasing?
a) 4.92 kph
b) 6.21 kph
c) 2.64 kph
d) 3.34 kph
d) 3.34 kph
A battleship is 1.5 km from a straight shore. It is targeting an enemy troop running along the shore at 4 kph. How fast is the gun of the battleship rotating when the troop is 500 m from the point on the shore nearest to the battleship in revolutions per hour?
a) 0.3820 rev/hr
b) 0.5722 rev/hr
c) 0.7965 rev/hr
d) 0.9283 rev/hr
a) 0.3820 rev/hr
The legs of a right triangle are 70 cm. If one of the legs starts to shrink at the rate of 5 cm/min, and the other increases at the same rate, how fast is the length of the hypotenuse of the triangle changing 2 min later?
a) 1 cm/min
b) 2 cm/min
c) 5 cm/min
d) 7 cm/min
a) 1 cm/min
There is a constant inflow of liquid into a conical vessel 15 ft deep and 7.50 ft in diameter at the top. Water is flowing at the rate of 6 ft³/min. When the water is 4 ft deep, determine the rate of water rise.
a) 1.68 ft/min
b) 1.78 ft/min
c) 1.84 ft/min
d) 1.91 ft/min
d) 1.91 ft/min
Water is flowing at constant rate of 125.664 cm³/s in a hemispherical bowl with radius 20 cm. If the height of water is increasing at 0.196 cm/s, what is the height of the water at this instant?
a) 4 cm
b) 5 cm
c) 6 cm
d) 7 cm
c) 6 cm
A spherical snowball is melting in such a way that its surface area decreases at the rate of 1 in²/min. How fast is the radius shrinking when it is 3 in?
a) 0.3655 in
b) 0.2144 in
c) 0.0133 in
d) 0.0041 in
c) 0.0133 in
Determine the radius of curvature at (4, 4) of the curve y² - 4x = 0.
a) 23.4
b) 22.4
c) 25.4
d) 24.4
b) 22.4
Find the area of circle tangent to y = x³ + 3x - 1 at (1, 3).
a) 3450
b) 4420
c) 4590
d) 5680
b) 4420
