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Derivatives

100

d/dx(x²)

2x

200

d/dx(5x⁴ + 18x³ + 2x² + 50x)

20x³ + 54x² + 4x

300

d/dx(-(x+1)/(4x+7))

-3/(4x+7)^2

400

d/dx(arcsin(5x²))

10x/sqrt(1-25x⁴)

500

d/dx((9x²-4x+9)/(2(x²-3)²sqrt(3x-1))

(-27x⁴+36x³-386x²+180x+57)/(4(x²-3)²(3x-1)sqrt(3x-1))

Limits

100

lim_x->2(x²-4)

0

200

lim_x->2((x²-x-2)/(x²-2x))

3/2 or 1.5

300

lim_x-2((x²-4)/(x-2))

4

400

lim_x->3((1/(x²-1))-(2/(x⁴-1)))

1/10 or 0.1

500

lim_x->-1((x³+x²+x+1)/(x⁴+x²-2))

-1/3 or -0.333

Integrals

100

int(x) dx

x²/2 + C

200

int(sin(x)) dx

-cos(x) + C

300

int(1/(x²+4)) dx

1/2arctan(x/2) + C

400

int(1-tan(x+2)) dx

x+ln|cos(x+2)| + C

500

int(1/(x-sqrt(1-x²))) dx

1/2ln|tan²(1/2arcsin(x))+2tan(1/2arcsin(x))-1|-1/2ln|tan²(1/2arcsin(x))+1|-arctan(tan(1/2arcsin(x)))+C

Applications of Derivatives/ Integrals

100

The relationship between m and v is defined by the equation 8x³=sqrt(4v). 

At the instant when dv/dt=12 and v=16, what is the value of dx/dt?

1/8

200

Let g be the function defined by g(x) = 5 + 4x − x³

What is the minimum value of the function on the interval [−2, 1] ?

1.921

300

Let h(x) = x + 3e^x−4

Use the equation of the tangent line to h(x) at x = 4 to approximate h(5).

11

400

Bob rides his bike along a straight path. Bob’s velocity can be modeled by the function

v(t) = (t + 2)e^sqrt(t) meters per minute for 0 ≤ t ≤ 10 minutes.

Find Bob’s acceleration at time t = 4. Indicate units of measure.

At time t=4 minutes, Bob's acceleration is 18.473 meters per minutes squared.

500

The derivative of the function f is f'(x) = (x³+x)²cos(2.5x)

On what interval(s) in (-1, 1) is the graph of f concave down?

(-0.466, 0) and (0.466, 1)

Random

100

d/dx(lnx²)

2/x

200

lim_x->inf.(-7x²-2x-13)

-inf.

300

int(xe^2x) dx

(xe^2x)/2-(e^2x)/4+C

400

f(x) = 4x²-12x, what is f''(x)?

f''(x)=8

500

limx->33((sqrt(x+3)-6)/(x-33))

1/12