Calculus I - Exam III Review

L'Hôpital's Rule

100

Evaluate:

lim⁡[x→0] (sin⁡(x) / x)

1

200

Evaluate:

lim⁡[x→∞] (x/e^x)

0

300

Evaluate:

lim⁡[x→π] ((x-π)/cos(x))

0

400

Evaluate:

lim⁡[x→1] ((ln(x))/(x-1))

1

500

Evaluate:

lim⁡[x→0⁺] (ln(x)/x)

-∞

MVT & Rolle’s Theorem

100

Find all c in (1,3) that satisfy the MVT in f(x)=x² on [1,3]

c =2

200

Find all c in (-1,3) that satisfy the MVT in f(x)=x²-2x-3 on [-1,3]

c = 1

300

Find all c in (1,e) that satisfy the MVT in f(x)=ln(x) on [1,e]

c = e-1

400

Find all c in (0,2π) that satisfy the MVT in f(x)=cos(x) on [0,2π]

c = π

500

Find all c in (1,4) that satisfy the MVT in f(x)=x³-3x²+5 on [1,4]

c = 1+√3

Extrema (FDT & SDT)

100

Find the critical points and classify them using the first derivative test for f(x)=x³-3x²+4

- At x=0, f(x) changes from increasing to decreasing: Local Maximum

- At x=2, f(x) changes from decreasing to increasing: Local Minimum

200

Find the critical points and classify them using the second derivative test for f(x)=x⁴-4x³+6x²

Since f′′(0) > 0, x=0 is a local minimum

300

Find the extrema for f(x)=x³-6x²+9x+5 using the first derivative test

- At x=1, f(x) changes from increasing to decreasing: Local Maximum

- At x=3, f(x) changes from decreasing to increasing: Local Minimum

400

Use the second derivative test to classify extrema for f(x)=2x³-9x²+12x+1

- At x=1, local maximum

- At x=2, local minimum

500

Identify absolute extrema for f(x)=-x³+6x²-9x+5 on [0,4]

- Absolute maximum: f(0)=f(3)=5

- Absolute minimum: f(1)=f(4)=1

Optimization

100

Find two numbers whose sum is 10 and whose product is as large as possible

x=5, y=5

200

A farmer uses 1600ft of fencing to enclose a rectangular area which will be divided into three pens of equal size. What is the maximum total area of the three pens that he can enclose with the limited amount of fencing that the farmer has available?

80,000 ft²

300

A box with a square base and an open top must have a volume of 32 cubic units. What dimensions minimize the surface area?

x=4, h=2

Thus, 4x4x2

400

Find the point P on the line y=4-x that is closest to the point (7,6)

P = (5/2, 3/2)

500

You currently sell a product for $20 and sell 100 units. For every $1 increase, you lose 4 customers. What would be the MAXIMUM revenue possible?

MAX Revenue = $2,025

- Price: $22.50

- Quantity (Q*) = 90 units

Antiderivatives

100

What is the integral of f(x) evaluated from 1 to 4, when

f(x) = 3x² - 2

57

200

What is the integral of f(x) evaluated from 0 to 1, when

f(x) = x(e^{x^2})

(e-1)/2

300

What is the integral of f(x) evaluated from 0 to 1/2, when

f(x) = 1/√(1-x²)

π/6

400

What is the integral of f(x) evaluated from 2 to 6, when

f(x) = 1/(x+1)ln(x+1)

ln(ln(7)) - ln(ln(3))

500

What is the integral of f(x) evaluated from 1 to 2, when

f(x) = 4x³/(1+x⁴)

ln(17/2)