Unit 4: Applications of the Derivative - Part 1
Unit 5: Applications of the Derivative - Part 2
Unit 6: Basic Integration and Applications
Unit 7: Advanced Integration and Applications
Wild Card!!!
100
f(x)=x^2+3/x^3
Find f''(x) .
f''(x)=2+36/x^5
100
Use the Extreme Value Theorem to determine the absolute extrema of f(x)=arctan(x^2) on [-2,1] .
f(x) is continuous on [-2,1], so the Extreme Value Theorem applies.
Absolute maximum: (-2, arctan4)
Absolute minimum: (0,0)
100
int(2x-4^x)dx=
x^2 - (4^x)/ln4 + C
100
Evaluate
int_1^5x/sqrt(2x-1)dx .
16/3
100
Evaluate lim_(x->oo) x^2/e^-x
lim_(x->oo) x^2/e^-x=0
200
f(x)=sin(x/2) on [0,4pi]
Give the intervals of f(x) where it is concave up and concave down. Justify your answer.
f(x) is concave up on (2pi,4pi) because f''(x)>0.
f(x) is concave down on (0,2pi) because f''(x)<0.
200
Determine the net and total distance traveled by the particle defined as x(t) from t=0 to t=5.
x(t)=t^3-6t^2+9t-2
Net distance = 20
Total distance = 28
200
A particle moves along the x-axis at a velocity of v(t)=1/(sqrtt), t>0 . At time t=1, its position is x=4. Find the acceleration and position functions for the particle.
x(t)=2sqrtt+2
a(t)=-1/(2sqrt(t^3))
200
Let g(x)=int_0^xf(t)dt , where f is the function whose graph is shown in the figure below.
Estimate g(8).
g(8)approx5
200
The base of a solid is bounded by the graphs y=x+1 and y=x^2-1. Cross sections of the solid perpendicular to the x-axis are rectangles of height 1. Find the volume of the solid.
V=9/2u^3
300
Given the curve xe^y-10x+3y=0
Find (dy)/dx.
(dy)/dx=(10-e^y)/(xe^y+3)
300
Determine whether the Mean Value Theorem (MVT) can be applied to f(x)=x^3+2x on the closed interval [-1,1] . If the MVT applies, find all values of c in the open interval (-1,1) such that
f'(c)=(f(b)-f(a))/(b-a) .
f(x) is continuous on [-1,1] and differentiable on (-1,1), so the MVT applies.
c=+-sqrt3/3
300
Approximate int_0^2x^3dx using a Trapezoidal Riemman Sum with four subintervals of equal length.
int_0^2x^3dx approx 17/4
300
Find the particular solution to the differential equation dy/dx=yxsinx^2 with the initial condition y(0)=1.
y=e^((1-cosx^2)/2)
300
Evaluate
int5/(x^2+3x-4)dx
lnabs((x-1)/(x+4))+C
400
A spherical balloon is inflated with gas at the rate 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 30 centimeters?
(dr)/dt=2/(9pi) (cm)/min
400
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 245,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing?
x = 700 m
y= 350 m
400
A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval 0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate W(t)=95sqrttsin^2(t/6) gallons per hour.
During the same time interval, water is removed from the tank at a rate R(t)=275sin^2(t/3) gallons/hour.
Write, but do not evaluate, an integral expression that finds how many gallons of water are in the tank at time t=18.
1200+int_0^18(W(t)-R(t))dt
400
Find the area of the region bounded by the two curves
f(y)=y^2 and g(y)=y+2
A=int_-1^2(y+2-y^2)dy
A=9/2
400
f(x)=-x^2+3x on [0, 3]
Determine if Rolle's Theorem can be applied to f on [0, 3]. If Rolle's Theorem can be applied, find all values of c guaranteed on (0, 3), such that f'(c)=0 .
Rolle's Theorem applies because f is a polynomial function and is, therefore, continuous on the closed interval, f' is a polynomial function and is, therefore, differentiable on the open interval, and f(0)=f(3)=0.
c=3/2 on (0, 3)
500
A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet away from the wall.
(d theta)/dt=1/12 (rad)/sec
500
A rectangle is bounded by the x-axis and the semicircle y=sqrt(25-x^2) . What length and width should the rectangle have so that its area is a maximum?
l=5sqrt2
w=(5sqrt2)/2
500
Find the average value of of f(x)=(4(x^2+1))/x^2 on the interval [1,3] and all of the values of x in the interval for which the function equals its average value.
Average Value= 16/3
x=sqrt3
500
Find the volume of the solid formed by revolving the region bounded by the graphs of y=sqrtx and y=x^2 about the x-axis.
V=(3pi)/10 u^3
500
Evaluate int((2x)/(x^2 +6x+13))dx
lnabs(x^2 +6x +13) -3arctan((x+3)/2) + C