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Displacement, Velocity, and Acceleration
Derivatives
Area/Volume
Limits and Continuity
Integration
100
x(t) is displacement. x(t)=3t^3 + 4t^2 + 3t + 2. Find the equation for velocity.
v(t)=9t^2 + 8t + 3
100
f(x)=4x^2 + 7x + 9. Find f''(x).
f"(x)=8
100
Find the area of the region enclosed by y=x^2 and y=x^0.5.
1/3
100
lim x -->2 (x^2 +7x+9)/(x^2 +13x+4)
27/34
100
Integrate: ∫x^2 + 6x +9dx
x^3/3 +3x^2 +9x +C
200
v(0)=3 v(1)=7 v(2)=9 v(3)=3 v(4)=6 Find the acceleration at 3 seconds
-1.5
200
State the physical interpretation of derivative.
Instantaneous Rate of Change
200
When rotated over the y-axis, find the volume of the region enclosed by the y-axis, y=1, and y=x^2/3. Leave answers in terms of pi.
pi/4
200
When is L'Hopital's rule used?
When a fraction is given and 0/0 is formed.
200
Integrate: ∫x^4+6x^3+2x^2+x+2dx
x^5/5 +3x^4/2 +2x^3/3 +x^2/2 +2x+C
300
If a(t)=x^2 - 10x +21, over what interval is the velocity decreasing?
(3,7)
300
f(x)=sin^-1(x^2). Find f'(x).
f'(x)=2x/(1-x^4)^1/2
300
Find the area of the region bounded by y=xe^-x^2, y=x+1, x=2, and the y-axis.
3.5092
300
State the three conditions needed for a function to be continuous
f(c) must exist lim x-->c must exist f(c) must equal lim x-->c
300
∫6x^-5 + 4x^2+6x^-4/3dx
(-3/2x^4)+(4x^3/3)-(18/x^1/3) +C
400
v(t)=-x^3 + 6x^2 -3x+7 Find the total distance traveled from x=0 to x=6.
99.306
400
f(x)=(tan(x^4)+1)^1/2
(2x^3)sec^2(x^4)/(tan(x^4)+1)^1/2
400
When rotated over the x-axis, find the volume of the region enclosed by the x-axis, x=1, x=4, and y=x^0.5. Leave answer in terms of pi
15pi/2
400
Find the horizontal asymptote. lim x-->inf (2x+3)(6x+1)(3x+4)/(3x+6)(2x+1)(2x+8)
y=3
400
∫sec^2(x)/9+tan^2(x)dx
tan^-1(x)/3 * (tan(x)/3)+C
500
a(t)=t^2 -50t+561 Find the maximum velocity of the particle given that v(0)=63
4012.667
500
f(x)=0.25(sinx)^2/(3x/5). Find f'(x)
f'(x)=(0.3xsinxcosx-0.15(sinx)^2)/0.36x^2
500
Rotated over y=-1, find the volume of the region enclosed by y=x^2 and y=x^0.5. Leave answer in terms of pi.
29pi/30
500
f(x)={x^3 - 4x^2 + kx+6 (-inf,2) {x^2-2x+5 [2,inf) What value of k makes the function continuous
7/2
500
∫5^xln(5) + 1/xln(9) -1/(1+81x/64)
5^x + log9x + cot^-1(9x/8) +C