UNIT 1 & 2
f(x)=
{ x2−1 if x<2
{3x-5 if x≥2
What does the limit equals at x=2?
A) 3
B) 1
C) 2
D) does not exist
What is does not exist, therefore not continuous
If the velocity of an object is increasing, this means the acceleration is ____ and the object is _____ over time.
A) positive, speeding up
B)negative, slowing down
C)negative, speeding up
D) positive, slowing down
What is option A) positive, speeding up
Let g be a twice-differentiable, increasing function of t. If g(0)=20 and g(10)=220, what must be true on the interval 0<t<10?
A) g'(t)=0 for some t in the interval
B) g'(t)=20 for some t in the interval
C) g''(t)=0 for some t in the interval
D) g''(t)>0 for all t in the interval
What is B) g'(t)=20 for some t in the interval
Which of the following is a solution to the differential equation 𝑥𝑦′−3𝑦=6 ?
A) y=x²+2
B) y=3x² - 2
C) y=5x³-2
D) y=7x³+2
What is option C, y=5x³-2?
Find the derivative of f(x)=3x2+7x-2 using the limit process.
A) 6x+7
B) 5x+3
C) 2x+1
D) 7x+6
What is 6x+7
As the limit approaches 0, what is the answer to the equation 1+sec^2(x)/cos(x)=
A) 0
B) 1
C) 2
D) 4
What is option C) 2
f(x) = 2x3-3x2-36x+2
Find the point of inflection(s)?
What is x = 1/2
The amount of bacteria in a petri dish increases at a rate proportional to the amount present. At time 𝑡=0, the amount of bacteria in the dish is 10 grams. At time 𝑡=2, the amount of bacteria in the dish is 30 grams. What is the amount of bacteria in the dish at time 𝑡=6 ?
A) 70 grams
B) 190 grams
C) 270 grams
D) 9270 grams
What is option C, 270 grams?
The limit of x approaching infinity is the square root of (9x4+1) / (x2-3x+5) =
A) 1
B) 3
C) 9
D) nonexistent
What is 3
If f(x)= sin2(2x+3), then f'(x)=
A) 4sin(2x+3)
B)2cos2(2x+3)
C)2sin(2x+3)cos(2x+3)
D)4sin(2x+3)cos(2x+3)
What is option D)4sin(2x+3)cos(2x+3)
The limit as n approaches infinity of the sum from k=1 to n of the quantity: the fourth root of 2+3k/n, multiplied by 3/n. Write this as a definite integral.
What is the integral from 2 to 5 of the fourth root of x dx.
Which of the following is the solution to the differential equation 𝑑𝑃𝑑𝑡+𝑃=10 with the initial condition 𝑃(0)=4 ?
A) P=-1+ the squareroot of (20t+25)
B) P=5-e-t
C) P=10- 6e-t
D) P=10-6et
What is option C, P=10-6e-t?
f(x) = { 2x-2 for x<3
{ 2x-4 for x>=3
For this piecewise function, which of the following is true?
I. lim (h->0^-) (f(3+h)-f(3))/h =2
II. lim (h->0^+) (f(3+h)-f(3))/h = 2
III. f'(3)=2
A) none
B) II only
C) I and II only
D) I, II, III
What is II only
f(x)= x2g(h(x)) where g(5)=3, g'(5)=-2, h(2)=5, and h'(2)=4. FIND f'(2).
A)-32
B)-20
C)0
D)4
What is option B) -20
Which of the following is equivalent to the integral of cos(4x)sin5(4x)dx?
A) ¼ integral cos udu multiplied by ¼ integral sin5udu, where u=4x
B) Integral cos(4x)dx multiplied by integral sin5(4x) dx
C) ¼ integral u5 du, where u=sin(4x)
D) Integral u5 du, where u=sin(4x)
What is C) ¼ integral u5 du, where u=sin(4x)
This integral represents the area between the curves y=sinx and y=cosx on the interval where cosx is greater than or equal too sinx:
What is the integral of (cosx-sinx) on the interval (0,pi/4)?
f(x) = x3sin(ln(x2+1))
f'(x) =
A) 3x2*sin(ln(x2+1))+(2x3/x2+1)*cos(ln(x2+1))
B) 3x2*sin(ln(x2+1))+(2x4/x2+1)*cos(ln(x2+1))
C) 3x2*sin(ln(x2+1))+(2x4/x2+1)*sin(ln(x2+1))
D) 3x2*sin(ln(x2+1))+(2x3/x2+1)*sin(ln(x2+1))
What is 3x2*sin(ln(x2+1))+(2x4/x2+1)*cos(ln(x2+1))
The edge of a cube is increasing at a rate of 0.2 inches per second. At the instant when the total surface area becomes 150 square inches, what is the rate of increase, in cubic inches per second, of the volume of the cube?
A)5
B)10
C)15
D)20
What is C) 15
A particle moves along a straight line with its velocity at time t, given by V(t)=t2-4t+3, where v(t) is measured in meters per second and t is measured in seconds for 0<=t<=5. The particle starts at s(0)=7 meters at t=0.
Determine the time intervals during which the particle is moving to the left and the right. Justify your answer.
Moving to the right between (0,1)U(3,5) because the velocity is positive.
Moving to the left between (1,3) because the velocity is negative.
A region is bounded by the curves y= sqrt of x, y=0, and x=4. Give the integral expression and then find the volume of the solid formed when this region is rotated about the line y=2.
What is pi times the integral of [(2)2-(2-sqrt of x)2]dx from b=4 and a=0? And what is (40pi)/3?