Calculus Review

Derivatives

100

Find the derivative of this equation:

y=3x⁴

Answer: 

12x³

200

find the derivative:

y=(2x+1)²

Answer: 

8x+4

300

find the derivative:

y= x²+x-2/(x³+6)

Answer:

(x³+6)(2+1)-(x²+x-2)(3x²)/(x³+6)²

400

find the derivative:

y=2secx-cscx

Answer: 

2secxtanx+cscxcotx

500

Find derivative:

cos(x²+2y)+xe^{y^2=1}

Answer:

(2xsin(x²+2y)-e^{y^2})/2yxe^{y^2}-2sin(x²+2y)

Integrals/antiderivatives

100

What is the integral of sin x?

Answer:

-cosx + C

200

Question:

Find the antiderivative of f(x)=3√(x)-2(³√x)

Answer:

F(x)=2x^3/2-(3/2x^4/3)+C

300

Question:

Find f if f'(x) = e^x+20(1+x²)^-1 and f(0)=-2

Answer:

f(x)=e^x+20tan^-1-3

400

Question: What is the value of

∫₁² x³+2x⁶/(x⁴)

Answer:

ln(2)+7

500

Question:

If f(x)= ∫₁^{x^3} (1/(1+lnt))dt for x≥1, then f'(2)=?

Answer:

12/(1+ln8)

Rates

100

Lynnette can wash 95 cars in 5 days. How many cars can Lynnette wash in 11 days?

209 cars

200

What is the formula for the volume of a sphere?

V=4/3πr²

300

A rectangle's length increases at a rate of 8 cm/sec, its width increases at a rate of 3cm/sec.  When the length is 20cm and the width is 10cm, how fast is the area of the rectangle increasing?

140 cm²/sec

400

The penguin population on an island is modeled by the differentiable function P(t) for 0 < t < 40, where t is in years. There are 100,000 penguins on the island at t = 0. The birth rate for the penguins is modeled by B(t) = 1000e0.06t penguins per year and the death rate is modeled by D(t) = 250e0.1t penguins per year. 

Is the penguin population increasing or decreasing at t = 5 years? Give a reason for your answer.

The penguin population is increasing at t=5 years because P(5) > 0 which means the birth rate is greater than the death rate

500

A 26 ft long ladder leans against a vertical wall. If the lower end is being moved away from the wall at the rate of 5ft/sec, how fast is the height of the top decreasing when the lower end is 10 ft from the wall?

dy/ dt = -100/48 = -2.083 ft/sec

U Substitution

100

Find the du. ∫3x²sin(x³)dx

du=3x²dx

200

What is the du of ∫(x)/sqrt(1-4x²)dx?

-1/8du=dx

300

Calculate ∫e⁵xdx.

1/5e^4+c

400

Evaluate ∫(x⁵)/(1+x¹²)dx. 

1/6arctan(x⁶)+c

500

Evaluate  ∫²₀ x²sqrt x³+1dx

52/9

Limits

100

How do you calculate average velocity?

distance traveled/time elapsed

200

Evaluate lim_x-->0(sinx)/(x)

1

300

Evaluate lim_x-->-2(x³+2x²-1)/(5-3x)

-1/11

400

Let f be the function defined by f(x) = sqrtx+1 for 0<x<3 and 5-x for 3<x<5. Is f continuous at x=3? 

*(<) means less than and equal to

1. f(3) = 2

2. lim_x-->3f(x)=2

3. lim_x-->3f(x)=f(3); 2=2

yes, f(x) is continuous at x=3

500

if 2x<g(x)<x⁴-x²+2 for all x, evaluate lim_x-->1g(x)

*(<) mean less than and equal to

Lim 2x=1

lim_x-->1x⁴-x²+2=2

Therefore, by squeeze theorem lim_x-->2g(x)=2