General Derivatives
d/dx xa = ?
= ax(a - 1)
f(x) = 20x + 4. What is the indefinite integral of f(x)?
10x2 + 4x + C
The position of an object is given by 4x2 + 3x - 2. Find the equation of the line tangent to the graph at x = 1.
y - 5 = 11(x - 1)
f(x) = 4x2 + 6x - 4.
f'(x) = ?
f'(x) = 8x + 6
What is the area between the curves f(x) = 3x2 + 4 and g(x) = x3 - 6x + 10? Round all values to 3 decimal places, from the bounds to the answer.
41.169 units2
d/dx (5x2)(sqrt(2x + 3)) = ?
= (25x2 + 30x)/sqrt(2x + 3)
OR [5x(5x + 6)]/(sqrt(2x + 3))
What is Integral (3->5) ((sqrt(3)/5)x2 - (5/2)x + 1) (dx)? Round to 3 decimal places.
-6.684
Use the line tangent to the graph of y = sqrt(x4 - 0.22) at x = 3 to approximate the value at x = 4. Round all values before usage to 3 decimal places.
Find f'(x) and f''(x) for the equation f(x) = 20x^4 + 9x^3 - 4x.
f'(x) = 80x^3 + 27x^2 - 4.
f''(x) = 240x^2 + 54x.
The cross-sections of an object whose base is encompassed by the functions y = (1/5)x3 - x2 + 10 and y = 2x2 - 5x are equilateral triangles perpendicular to the x-axis.
Find the volume of this object. Round to the nearest thousandth; being a little off is mostly fine!
310.790 units3
d/dx sqrt(3x2)/9x3 = ?
= -(2sqrt(3))/9x3
What is the indefinite integral of 1/sqrt(1 - x2)?
arcsin(x)
The rate at which water is filling up a water tanker over time is given by the equation r(t) = 4t3 + 2t - 9. If the initial amount of water in the tanker was 20 liters, give an equation for the amount of water in the tanker and the increase in the rate of water flow, using w(t) and a(t) for the respective equations, then find the rate of water flow, the amount of water in the thanker and the rate of increase in the water flow at t = 3.
w(t) = t4 + t2 - 9t + 20
a(t) = 12t2 + 2
w(10) = 83 liters
r(10) = 105 liters/sec
a(10) = 110 liters/sec2
f''(x) = 20x - 12. If f'(x) = 6 at x = 1 and f(x) = 4.25 at x = 1.5, find the equation for f(x).
(10/3)x^2 - 2x^2 + x - 4
The area bound by the functions (1/5)x2 + 5, y = 10 and the y-axis is rotated around the x-axis.
What is the volume of the resulting shape?
800pi/3 units3
d/dx 3e3x + 5ln(15x3) = ?
= 5/x + 9e3x
What is d/dx Integral (3 -> x2) (5t3 - (2/9)t2 + 1)dt?
5x6 - (2/9)x4 + 1
The acceleration of an object is given by the function a(x) = (4/3)x + (3/4). If the object's velocity is 7.35 at x = 3 and the object's position is approximately 3.478 at x = 2, find the equations for the object's velocity and position, using v(x) and p(x) for the respective equations.
v(x) = (2/3)x2 + (3/4)x - 9/10
p(x) = (2/9)x3 + (3/8)x2 - (9/10)x + 2
If f'(x) = 3x2 + 4x - 10 and f(x) at x = 2 is equal to 4, what is f'(x) + 2f(x) - 3f''(x)?
= 2x3 + 7x2 - 34x - 6
The cross-sections of an object whose base is encompassed by the functions y = (1/5)x3 - x2 + 10 and y = 2x2 - 5x are equilateral triangles perpendicular to the x-axis.
Find the volume of this object. Round to 3 decimal places. Being slightly off is mostly fine!
310.790 units3
d/dx [ex + 2sqrt(3x)]/[5 + ln(x)] = ?
= [xex ln(x) + 5xex + 31/2x1/2ln(x) + 33/2x1/2 - ex]/[x(5 + ln(x))2]
What is Integral (-7 -> -1) (3e5x - (1/5)x2 + 3)dx? Round to 3 decimal places.
-4.796
The volume of a cube is increasing at a certain rate R. The length of the same cube is increasing at a rate of 3.5 m/sec. Find R when the length of the cube is 7 m.
514.5 m3/sec.
f(x) = (5x6 + 7x2)/(3x4 + 1) What will f''(x) be?
f''(x) = 2(45x12 + 189x8 - 177x4 + 7)/(3x4 + 1)3
The area bound by y = (1/5)x3, y = -(2/3)x2 + 6 and the y-axis is rotated around the line y = -2.
Find the volume of the resulting solid. Round to 3 decimal places.