Calculus I - Exam II Review

Derivatives

100

f(x) = x²⋅sin(x)

Find f'(x).

2xsin(x) + x²cos(x)

200

f(x) = (3x²+1)/x

Find f'(x).

(3x²-1)/x²

300

f(x) = (2x³+5)⁴

Find f'(x).

24x²(2x³+5)³

400

f(x) = tan(x²)

Find f'(x).

2x⋅sec²(x²)

500

f(x) = ln(x²+1)⋅e^3x 

Find f'(x).

2x/(x²+1)⋅e^3x + ln(x²+1)⋅3e^3x

ImplicitDiff.

100

Find dy/dx.

x²+y²=25

-x/y

200

Find dy/dx.

y² = x³+y

3x²/(2y-1)

300

Find dy/dx.

sin(y)=x²+2y

2x/cos(y)-2

400

Find dy/dx.

e^y=x²+y³ 

2x/(e^y-3y²)

500

Find dy/dx.

cos(xy)=x+y 

[y⋅sin(xy)+1]/-(x⋅sin(xy)+1)

Related Rates

100

A square metal plate is heated, causing its side length to expand at a rate of 0.02 cm/min. At the moment when the side length is 10 cm, how fast is the area of the plate changing?

The area of the plate is increasing at 0.4 cm²/min.

200

A circle's radius is increasing at a rate of 2 cm/s. Find the rate at which the circle's area is changing when the radius is 5 cm.

The area is increasing at 20π cm²/s.

300

Air is being pumped into a spherical balloon at a rate of 100 cm³/s. Find the rate at which the radius is changing when the radius is 10 cm.

The radius is increasing at 1/4π cm/s.

400

A ladder 10 ft long is leaning against a wall. If the bottom of the ladder slides away from the wall at 2 ft/s, how fast is the top of the ladder sliding up or down the wall when the bottom is 6 ft from the wall?

The top of the ladder is sliding down at 3/2 ft/s

500

A conical tank is being filled with water at a rate of 5 m³/min. The tank has a height of 12 m and a radius of 4 m. Find the rate at which the water level rises when the water is 3 m deep.

The water level is rising at 5/π m/min.

Linearity

100

Find the linear approximation FORMULA [L(x)] of f(x)=sqrt(4) at x=4

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

200

Use linearization to approximate sqrt(49.5)

 = 7.036 (approx)

300

Find the linear approximation of f(x)=e^x/2 at x=0, then use it to approximate e^0.1

= 1.05 (approx)

400

Approximate cos⁡(0.1) using the linearization of f(x)=cos⁡(x) at x=0

= 1 (approx)

500

The temperature T (in °C) of a metal rod t minutes after being placed in a room follows the equation T(t)=150e^-0.2t+20. Use linearization at t=0 to approximate the temperature after 1 minute

T(1) = 140 C (approx)

Inverse Functions & Their Derivatives

100

f(x)=2x+7; find f^-1(x)

f^-1(x) = (x-7)/2

200

f(x)=3e^2x−6; find f^-1(x)

f^-1(x) = 1/2[ln((x+6)/3)]

300

Let f(x)=ln⁡(x+5). Find (f⁻¹)′(0)

(f⁻¹)′(0) = 1

400

Let f(x)=x³+x. Find (f⁻¹)′(2)

(f⁻¹)′(2) = 1/4

500

Let f(x)=x+e^x. Find (f⁻¹)′(1+e)

(f⁻¹)′(1+e) = 1/(1+e)