Midterm 1 Review!

Related Rates

100

A spherical balloon is being filled with air at the constant rate of 2 cm³/sec. How fast is the radius increasing when the radius is 3cm?

r'(t) = 1/18pi cm/sec

200

An airplane is flying overhead at a constant elevation of 4000ft. A man is viewing the plane from a position 3000ft from the base of a radio tower. The airplane is flying horizontally away from the man. If the plane is flying at the rate of 600ft/sec, at what rate is the distance between the man and the plane increasing when the plane passes over the radio tower?

ds/dt = 0 

300

A rocket is launched so that it rises vertically. A camera is positioned 5000ft from the launch pad. When the rocket is 1000ft above the launch pad, its velocity is 600ft/sec. Find the necessary rate of change of the camera’s angle as a function of time so that it stays focused on the rocket.

d(theta)/dt = 3/26 rad/sec

400

Water is draining from the bottom of a cone-shaped funnel at the rate of 0.03ft³/sec. The height of the funnel is 2 ft and the radius at the top of the funnel is 1ft. At what rate is the height of the water in the funnel changing when the height of the water is 1/2 ft?

dh/dt = -.48/pi = -.153 ft/sec

500

find dz/dt at (x, y) = (1, 3) and z² = x² + y² if dx/dt = 4 and dy/dt = 3

13/sqrt(10)

Linear Approximation

100

When is linear approximation constant? 

f'(x) = 0

200

When is a linear approximation exact?

When y = f(x) is linear or constant

300

f(x) = sin²x at a = 0

L(x) = 0

400

f(x) = 1/x at a=2

L(x) = 1/2 - 1/4(x - 2)

500

What is the definition of Linear Approximation?

L(x) = f(a) - f'(a)*(x-a)

Differentials

100

find the differential of y = xcosx

𝑑𝑦=(cos𝑥−𝑥sin𝑥)𝑑𝑥

200

dV if a circular cylinder of height 3 changes from 𝑟=2 to 𝑟=1.9cm.

-1.2 pi cm³

300

A pool has a rectangular base of 10 ft by 20 ft and a depth of 6 ft. What is the change in volume if you only fill it up to 5.5 ft?

-100 ft³

400

𝑦=𝑥³+2𝑥+1/𝑥

x=1

𝑑𝑥=0.05

compute dy using differentials

.2

500

Suppose the side length of a cube is measured to be 5 cm with an accuracy of 0.1 cm. Use differentials to estimate the error in the computed volume of the cube. 

dV <= 7.5 

Maxima and Minima

100

Find absolute max and min of:

𝑓(𝑥)=𝑥²−3𝑥^2/3  over [0,2].

max: f(0) = 0

min: f(1) = -2

200

Find the absolute max and min of:

𝑓(𝑥)=−𝑥²+3𝑥−2 over [1,3].

max: f(3/2) = 1/4

min: f(3) = -2

300

Define a critical point

a point c is a critical point if f'(c) = 0 or if f'(c) is undefined

400

Find absolute max and min

𝑦=𝑥²+2/𝑥 over [1,4]

max: f(4) = 33/2 

min: f(1) = 3

500

Find local min and max:

y = (x² + x + 6) / (x - 1)

local max: 

f(1 - 2sqrt(2)) = 3 - 4sqrt(2) 

local min:

f(1 + 2sqrt(2)) = 3 + 4sqrt(2)

Derivatives and Shapes of Graphs

100

What conditions mean a graph is concave up or concave down?

Concave up: first derivate is increasing and second derivative is positive

concave down: first derivative is decreasing and second derivative is negative

200

What do the following facts tell you about a local extrema, c:

- f''(c) > 0

- f''(c) < 0

-f''(c) = 0

- f has a local minimum at 𝑐.

- f has a local maximum at 𝑐.

- inconclusive

300

List all inflection points of:

f(x) = x³ - 6x² +9x + 30

(2, 32)

400

Find the intervals where f is concave up and concave down:

f(x) = x³ - 6x²

Concave up for 𝑥>2, concave down for 𝑥<2

500

find: 

1. intervals where 𝑓f is increasing or decreasing,
2. local minima and maxima of 𝑓,f,
3. intervals where 𝑓f is concave up and concave down, and
4. the inflection points of 𝑓.

f(x) = 1 + x + x²

a. Increasing over x>−1/2, decreasing over x<−1/2 b. Minimum at x=−1/2 c. Concave up for all x d. No inflection points