Optimization
A particle's position is given by the function x(t)=t^3+t^2-16t
What is the furthest right the particle reaches on the interval [0,5]?
The particle reaches a maximum position of x=70 at t=5.
For a function whose values are shown in the following table, how many subintervals would the Riemann sum have?
| x | 0 | 2 | 4 | 6 | 8 | 10 |
| y | 5 | 7 | 3 | 12 | 15 | 0 |
5 subintervals
Solve:
int (3x^2 + 4x + 3) dx
What is:
x^3 + 2x^2 + 3x + C
Integrate:
dy/dx = -sin(5x)
What is:
y=1/5cos(5x)+C
Integrate:
dy/dx = cos(3x)
What is:
y = 1/3sin(3x)+C
A rectangular field needs to enclose 169 square yards of area. Find the dimensions of the field that will require the minimum amount of fencing.
13 yards x 13 yards
Use a left Riemann sum with 3 subintervals to calculate the area under the curve given by the table below.
| x | 0 | 3 | 6 | 9 |
| y | 10 | 15 | 17 | 16 |
A=126
Find the equation f(x), if f'(x) = 6x and f(0)=8.
What is: f(x)=
3x^2 + 8
Find f(x):
int(2x^5(x^6 + 2)^3 dx
What is: f(x)=
1/12 (x^6 +2)^4 + C
Integrate:
int (x^2)/(sqrt(x^3 + 3))dx
What is:
2/3 sqrt (x^3 + 3) + C ?
A particle's position is given by the function x(t)=-t^3+15t^2-49t
What is the maximum velocity the particle reaches on the interval [0,7]?
The particle reaches a maximum velocity of 124 at t=5.
Use a right Riemann sum with 5 subintervals to estimate the area under the curve y=x2 on the interval [0,5].
A=55
Find the integral:
int (sqrtx+ 1/(2sqrtx))dx
What is:
Find the integral:
int_-2^1 6x(5-2x^2)^2 dx
-27
Integrate:
int_0^(pi/3) (sin^3 x cos x)dx
What is:
9/64
Find the area of the rectangle of largest area that has its base on the x-axis and its other 2 vertices lying above the x-axis on the parabola y=-2x^2+216.
1728 square units
Values of f(x) are given below. Use a left Riemann sum with 4 subintervals to estimate the area under the curve.
| x | 0 | 3 | 5 | 9 | 15 |
| f(x) | 12 | 11 | 5 | 7 | 8 |
A=120
Integrate:
int_0^(pi/3) sec^2 x
What is:
sqrt 3
Find f(x) if
dy/dx = x/(4 + x^2)^(1/2)
What is f(x)=
(4 + x^2)^(1/2) + C
Integrate:
dy/dx= (sin(1/x))/(6x^2) dx
What is: f(x)=
1/6 cos(1/x) + C
588 square centimeters of material is available to make a box with a square base and an open top. Find the largest possible volume of the box.
1372 cubic cm
A function f(x) is always increasing. Will using a left Riemann sum overestimate or underestimate the area under the curve?
Underestimate
Find f(x), given: f''(x) = sin x, f'(0)= 2, f(0)=8.
What is: f(x)= -sinx + 3x + 8?
Integrate:
dy/dx = x(x+2)^(1/2)
What is:
2/5(x+2)^(5/2) - 4/3(x+2)^(3/2) + C
Let f be a function such that int_6^15 f(u)du=6. What must be the value of int_-6^3 xf(1/3x^2+3)dx?
-9