Unit 5-6

Find all critical numbers for the function: 𝑓(𝑥) = (9 − 𝑥2)3/5

(-3,0,3)

Find the sum of the values of 𝑎𝑎 and 𝑏𝑏 such that 𝑓𝑓(𝑥𝑥) = 2𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑏𝑏 + 3 has a relative extremum at the point (−1, 2).

5/2

f3-1 d/dt (2t^2+3t-1)dt

58

What are all values of 𝑥𝑥 for which the function 𝑓𝑓 defined by 𝑓(𝑥) = (𝑥2 − 3)𝑒−𝑥 is increasing?

-1<x<3

The graph of 𝑓𝑓(𝑥𝑥) = 𝑥^4 + 8𝑥^3 − 72𝑥^2 + 6 is concave down for

-6<x<2

f x/sqrt 4-x^2

-sqrt 4-x^2+C

S(2+lnx)^3 / x dx

1/4 (2_lnx)^4+C

fx/x^4+16dx

1/8 tan-1(x^2/4)+c

A tank contains 120 gallons of oil initially. Oil is being pumped out of the tank at a rate 𝑅(𝑡), where 𝑅(𝑡) is measured in gallons per hour, and 𝑡𝑡 is measured in hours. The table below shows selected values for 𝑅(𝑡). Using a trapezoidal approximation with three subintervals and the data from the table, find an estimate of the number of gallons of oil that are in the tank at time 𝑡 = 14 hours.

100.8

A right circular cylinder is inscribed inside a sphere with radius 𝑟𝑟 = 3 inches. Find the maximum volume of the right circular cylinder. Volume of cylinder: 𝑉𝑉 = 𝜋𝜋𝑟𝑟2ℎ .

12sqrt3

A particle is moving along the x-axis with a velocity of 𝑣𝑣(𝑡𝑡) = 8 cos 1/2. The particle is at 𝑥 = 2 when 𝑡 = 0. Find the position of the particle when 𝑡 = 𝜋.

18

The graph of the second derivative of a function 𝑓𝑓(𝑥𝑥) is shown at right. Which of the following is/are true?

The graph of 𝑓𝑓(𝑥𝑥) has a point of inflection at 𝑥𝑥 = - 1. II. The graph of 𝑓𝑓(𝑥𝑥) is concave down on the interval (- 1, 3). III. The graph 𝑓𝑓 ′ (𝑥𝑥) is decreasing at 𝑥𝑥 = 2.

Find the values of 𝑥𝑥 where the graph of the function has a point of inflection. Given: 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 − 𝑥𝑥3

X=0, x=1/2

f e^ax+b dx

e(ax/2+bc+c)+C

If the substitution 𝑢𝑢 = √𝑥 + 1 is made, the integral f8-3 sqrt x+1 /x dx

f3-2 2u^2/u^2-1

f 2^x/3dx

ln 3(2^x/3)/2 +C

Consider the function 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 + 6𝑥𝑥2 − 36𝑥𝑥 + 4. Then

𝑓𝑓 has a local minimum at 𝑥𝑥 = 2 III. 𝑓𝑓 has local maximum at 𝑥𝑥 = −6