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Derivatives Limits Integrals

Applications Misc.

Other stuff

$100 $100 $100 $100 $100 $100

$200 $200 $200 $200 $200 $200

$300 $300 $300 $300 $300 $300

$400 $400 $400 $400 $400 $400

$500 $500 $500 $500 $500 $500

Final Jeopardy

Scores

$100$100

Calculate the derivative of

$100$100

$200$200

f(x) = x sin (x)

$200$200

f’(x) = x cos (x) + sin (x)f’(x) = x cos (x) + sin (x)

$300$300

Y = X2(X- 2)5

$300$300

y’ = 5x2(x-2)4 + (x-2)5(2x)

y’ = x(x-2)4(7x-4)

y’ = 5x2(x-2)4 + (x-2)5(2x)

y’ = x(x-2)4(7x-4)

$400$400

Use Implicit Differentiation to calculate y’

X2 + XY + Y3 = 3

Use Implicit Differentiation to calculate y’

X2 + XY + Y3 = 3

$400$400

2x + xy’ + y + 3y2 y’ = 0

y’ = -2x – y

X + 3y2

2x + xy’ + y + 3y2 y’ = 0

y’ = -2x – y

X + 3y2

$500$500

Find the derivative of the following function

F(x) = x3 – 3x2 + 4

Find the derivative of the following function

F(x) = x3 – 3x2 + 4

$500$500

y’ = x3-8

$100$100

Evaluate

Lim 6X - 3

Evaluate

Lim 6X - 3X → 4X → 4

$100$100

$200$200

Evaluate

Lim sin x

Evaluate

Lim sin x

xX → 0X → 0

$200$200

$300$300

Evaluate

Lim l x – 4 l

Evaluate

Lim l x – 4 l

x - 4X → 4X → 4

$300$300

No limitNo limit

$400$400

Evaluate

Lim X2 – 9

Evaluate

Lim X2 – 9

X + 3X → - 3X → - 3

$400$400

$500$500

Evaluate

Lim (X + ΔX)2 + 1 – (X2 + 1)

Evaluate

Lim (X + ΔX)2 + 1 – (X2 + 1)

ΔXΔX → 0ΔX → 0

$500$500

(X+ΔX)^2 + 1 – X^1 -1 Δ X cancel 1’s

X^2 + 2XΔX + ΔX^2 – X^2 Δ X square and simplify

2X + Δ XΔ X = 0Answer = 2X.

(X+ΔX)^2 + 1 – X^1 -1 Δ X cancel 1’s

X^2 + 2XΔX + ΔX^2 – X^2 Δ X square and simplify

2X + Δ XΔ X = 0Answer = 2X.

$100$100

Evaluate

∫ 2x dx

Evaluate

∫ 2x dx

$100$100

x2 + cx2 + c

$200$200

Evaluate

∫ x3 + 2x

Evaluate

∫ x3 + 2x

$200$200

1/3 x3 +2x + c

(cancel one x and then integrate)

1/3 x3 +2x + c

(cancel one x and then integrate)Home

$300$300

Evaluate

∫ (lnx)3 dx

Evaluate

∫ (lnx)3 dx

$300$300

Let U = Lnx, du = 1/x dx

= ¼ (ln x)4 + c

Let U = Lnx, du = 1/x dx

= ¼ (ln x)4 + c

$400$400

Evaluate

∫ cos 2x dx

Evaluate

∫ cos 2x dx

$400$400

½ sin(2x) + c½ sin(2x) + c

$500$500

Evaluate

∫xe-x2 dx

Evaluate

∫xe-x2 dx

$500$500

Let u = -x2, du = -2x

-½ ∫ eu du

-½ e-x2+c

Let u = -x2, du = -2x

-½ ∫ eu du

-½ e-x2+c

$100$100

In an application problem relevant to motion, the first

derivative of an equation tells ____________.

In an application problem relevant to motion, the first

derivative of an equation tells ____________.

$100$100

VelocityVelocity

$200$200

In an application problem relevant to motion, the second

derivative tells ______________.

In an application problem relevant to motion, the second

derivative tells ______________.

$200$200

AccelerationAcceleration

$300$300

A hot-air balloon is rising straight up from a level field and is being tracked by a range finder located

500ft from the balloon. At the moment the range finder’s angle of elevation is π/4, the angle is increasing at the rate of 0.14

rad/min. How fast is the balloon rising at that moment?

A hot-air balloon is rising straight up from a level field and is being tracked by a range finder located

500ft from the balloon. At the moment the range finder’s angle of elevation is π/4, the angle is increasing at the rate of 0.14

rad/min. How fast is the balloon rising at that moment?

$300$300 Draw a picture

Θ = the angle the range finder makes with the ground

Y = the height of the balloon (in feet)

T = time, and θ and y are differentiable functions of t.

Y = 500 tan θ

Y’ = 500 (sec2θ)dθ/dt

Y’ = 500(2)(.14) = 140ft/min

Draw a picture

Θ = the angle the range finder makes with the ground

Y = the height of the balloon (in feet)

T = time, and θ and y are differentiable functions of t.

Y = 500 tan θ

Y’ = 500 (sec2θ)dθ/dt

Y’ = 500(2)(.14) = 140ft/minHome

$400$400

Air is being pumped into a spherical balloon to cause its

volume to increase at a rate of 100cm3/s. How fast is the radius of the balloon increasing when the

diameter is 50 cm?

Air is being pumped into a spherical balloon to cause its

volume to increase at a rate of 100cm3/s. How fast is the radius of the balloon increasing when the

diameter is 50 cm?

$400$400

1/(25π) cm/s1/(25π) cm/s

$500$500

Car A is traveling west at 50 mi/h while car B is traveling north at 60 mi/h. Both cars are headed

towards intersection C. At what rate are the cars approaching

each other when car A is 0.3 mi from the intersection, and car B is 0.4 mi from the intersection?

Car A is traveling west at 50 mi/h while car B is traveling north at 60 mi/h. Both cars are headed

towards intersection C. At what rate are the cars approaching

each other when car A is 0.3 mi from the intersection, and car B is 0.4 mi from the intersection?

$500$500

- 78 mi/h- 78 mi/h

$100$100

Find the derivative of

y = ln(x3 + 1)

$100$100

y’ = 3x2

$200$200

y = 6(x2)

$200$200

y’ = (ln 6)(6^x^2)(2x)y’ = (ln 6)(6^x^2)(2x)

$300$300

Find the second derivative of

y = x3 + 3x2 – ½ x + 5

Find the second derivative of

y = x3 + 3x2 – ½ x + 5

$300$300

y = x3 + 3x2 – ½ x + 5

y’ = 3x2 + 6x – ½

y” = 6x + 6

y = x3 + 3x2 – ½ x + 5

y’ = 3x2 + 6x – ½

y” = 6x + 6

$400$400

Find y’

y = ln [x(1+x)2]

Find y’

y = ln [x(1+x)2]

$400$400

y’ = lnx +2ln(1+x)

y’ = 1/x +2/(1+x)

y’ = lnx +2ln(1+x)

y’ = 1/x +2/(1+x)

$500$500

A farmer has 2400 feet of fencing to create a rectangular pen for his

lamas next to their new barn. If the barn side off the pen does not

need a fence, what are the dimensions of the pen with the

largest area?

A farmer has 2400 feet of fencing to create a rectangular pen for his

lamas next to their new barn. If the barn side off the pen does not

need a fence, what are the dimensions of the pen with the

largest area?

$500$500A = LW = xy

A = 2x + y = 2400

y = 2400 - 2x

A = x(2400 – 2x) = 2400x – 2x2

A’ = 2400 – 4x

X = 600 ft

Y = 1200 ft

A = LW = xy

A = 2x + y = 2400

y = 2400 - 2x

A = x(2400 – 2x) = 2400x – 2x2

A’ = 2400 – 4x

X = 600 ft

Y = 1200 ftHome

$100$100

List the antiderivatives for the following trig functions

Sin Cos Tan

List the antiderivatives for the following trig functions

Sin Cos Tan

$100$100

-cos, sin, sec2-cos, sin, sec2

$200$200

Evaluate the integral

∫ (4 + 3x2) dx

Evaluate the integral

∫ (4 + 3x2) dxo

$200$200

$300$300

When is it good to use implicit differentiation?

$300$300

When an equation can not be easily solved for y in terms of x.

$400$400

Complete the statement.

Lim [ f(x) + c(g(x))] =

Lim f(x) + ________________

Complete the statement.

Lim [ f(x) + c(g(x))] =

Lim f(x) + ________________x → a

x → a

$400$400

+ c Lim g(x)+ c Lim g(x)

x → a

$500$500

In words, state the product rule and the quotient rule.

$500$500Product rule: first time the

derivative of the second + second time the derivative of the first.

Quotient rule: bottom time the derivative of the top – top times the derivative of the bottom all divided by the bottom squared.

Product rule: first time the derivative of the second + second

time the derivative of the first.

Quotient rule: bottom time the derivative of the top – top times the derivative of the bottom all divided by the bottom squared.

CalculusCalculus

Final Jeopardy Question

y = 2π4

y’ = 0y’ = 0

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