Test 2

(cotx)'

(-cscx)²

[((x²)+1))((x³)+3x+1)]

(5x⁴)+(12x²)+2x+3

X^x

(X^x)lnx + xX^x-1

(a^x)'

(a^x)×ln(a)

[(e^x)cosx ÷ cos2x]'

(e^x)(cosxcos2x - sinxcos2x + 2cosxsin2x) ÷ (cos2x)²

Quotient rule

(f(x)'g(x) - f(x)g(x)') ÷ g(x)²

dz/dx =

(g°f)'(x)

chain rule: prime

(g°f)(x) = g'(f(x))f'(x)

(x^m⁺n)'

(m+n)x^m+n-1

(tanx)'

(secx)²

(x^¹/x)'

(x^¹/x)[(1-lnx)/x²)

(arccotx)'

-1 / x²+1

(arccosx)'

-1 / √(1-x²)

(1 ÷ e^x)'

-1 ÷ e^x

((x+1) ÷ (x-1))'

-2 ÷ (x-1)²

(cos²(x))'

-2sinxcosx

(1 ÷ (x²)+1)'

-2x ÷ ((x²)+1)²

(cos(x²))'

-2xsin(x²)

There is a ladder against a wall with height = 4' base = 3' hypotenuse = 5' The base is moving out at a rate of 6in/sec How fast is the top of the ladder descending?

-4.5 in/sec

A private detective is following Mr Brown. They were walking along chestnut street, which is 30' wide. The detective has been following at a distance of 50'. Brown at the intersection with 4th ave, crossed 4th and then crossed chestnut (4th is 40' wide) The detective wants to keep Brown in sight, but from as far back as possible. Brown is moving at 4ft/sec. When he is 15' north of the intersection. How fast must the detective move?

-4/3 ft/sec

A boat is moored to a dock by a rope the lake is 2' lower than the dock. Someone on the dock is pulling in the rope, at 2 ft/min. When the remaining rope is 10' how fast is the boat approaching the dock?

-5/√6

(cosx)''

-cosx

(sinx)'''

-cosx

(sinx)¹¹⁵

-cosx

(cscx)'

-cotxcscx

(1 ÷ f(x))'

-f(x) ÷ f(x)²

(cos(cos(cosx)))'

-sin(cos(cosx)×-sin(cosx)×-sinx

(cosx)'

-sinx

(cosx)⁵

-sinx

(sinx)''

-sinx

lim h→0 (sin7h) ÷ 7

1

(arctanx)'

1 / x²+1

(arcsecx)'

1 / x√(x²-1) x>1 -1 / x√(x²-1) x<1

(arcsinx)'

1 / √(1-x²)

Find the linearization of y=√x near (1,1) and estimate √1.1 using the linearization

1+ ¹/₂(x-1) 1.05

Steps to find Related Rates

1. Identify variables ( given the rate of change for some of them and asked about the rate of change of others) 2.Identify relationships 3. differentiate to get a relation among the rates 4. solve for the desired rate

(f°g)'(x) set equal to g'(x)

1/ f'(g(x))

Air is being pumped into a balloon at 100cc/sec. When the radius is 50 cm how fast is the radius increasing?

1/100π cm/sec

A(triangle)

1/2 A(rectangle)

V(cone)

1/3 V(cylinder)

(ln(x))'

1/x

lim h→0 (tan2h) ÷ (tanh)

2

(tan2x)'

2(secx)²

(³√(x²+2x+2))'

2/3 [(x+1)/(x²+2x+2)²/³]

(e²x)'

2e²x

Double angle formula sin2x =

2sinxcosx

(tan(x²))'

2xsec²(x²)

Circumference

2πr

lim h→0 (sin3h) ÷ (sin5h)

3/5

The population of western elbonia in 1990 was 25 million. In 2000, it was 30 million. What would you expect the population to be in 2010?

36million

(e^3x)'

3e³x

SA(sphere)

4πr²

A chilled soda at 40°F is brought into a room at 70°F. After 10 minutes it's temp has risen to 45°F. What is the temp after ¹/₂ hours?

53°F

y(0)=5 y'(0)=10 what is y?

5e^(2t)

A conical paper cup with a height 5'' and diameter 2'' is being filled with water dribbling in at 2in³/sec. When the water halfway up the cup, how fast is it rising?

8/π in/sec

A population of bacteria has 50,000 members at 1:00 and 150,000 at 2:00. How many were there at 1:30?

86,600

Double angle formulas cos2x =

=(cosx)² - (sinx)² =2(cosx)² - 1 =1 - 2(sinx)²

Particle position is given by f(t)= t³-8t²+12t A. velocity at t=0? B. velocity after 1 second C. when is the particle at rest D. When is the particle moving in the positive direction E. find the total distance moved in the first 6 seconds G. find the acceleration at time T and after 1 second H. when is it speeding up or slowing down

A. 3t²-16t+12 B. -1 m/s C. (8±2√7)/3 D. before (8-2√7)/3, after (8+2√7)/3 E. forward (0, (8-2√7)/3), backward ((8-2√7)/3, (8+2√7)/3), forward ((8+2√7)/3, 6) G. f''(t)= 6t-16, f''(1)=-10 H. speeding up ((8-2√7)/3, ⁸/₃) ∪ ((8+2√7)/3, 6), slowing down (0, (8-2√7)/3) ∪ (⁸/₃, (8+2√7)/3)

[(e^x) ÷x]'

[(e^x)(x-1)] ÷ (x²)

[((x³)-1) ÷ ((x²)+x)]'

[(x⁴)+(2x³)+2x+1] ÷ [(x²)+x]²

Speed is...

absolute value of velocity

(cosx)⁴

cosx

(sinx)'

cosx

(sinx)⁵

cosx

(ln |sinx|)'

cotx

chain rule: leibniz

dz/dx = dz/dy × dy/dx

(xe^x)'

e^x(1-x)

decreased speed

f' > 0, f'' < 0 f' < 0, f'' > 0 different sign

increased speed

f' > 0, f'' > 0 f' < 0, f'' < 0 same sign

decreased velcoity

f'' < 0

increased velocity

f'' > 0

dy/dx =

f'(x)

(ln(f(x))'

f'(x) / f(x)

dz/dy =

g'(y)

y' =

ky

The tan line is the______ of the curve at that point

linearization

If T is the temp of a heated/cooled object where the ambient temp is T₀, T' is...

proportional to T-T₀

(r^x)'

r^x × lnr

Velocity is...

rate of change of position

Acceleration is...

rate of change of velocity

(x^r)'

rx^(r-1)

(cosx)'''

sinx

(sinx)⁴

sinx

cosy → ±√(1-x²)

sin²y+cos²y=1 x²+cos²y=1 cos²y=1-x²

(ln |secx|)'

tanx

(secx)'

tanxsecx

sinx + cosy = x²-y² I.D. with respect to y

x'= (siny -2y) / (cosx -2x)

y=√[(x²)+1] y'?

x/ (√([x²]+1))

second derivative x³+x = y³-y

y'= (3x²+1)/(3y²-1) y''= 6x(3y²-1)-6y(3x²+1)² / (3y²-1)³

sinx + cosy = x²-y² I.D. with respect to x

y'= (cosx - 2x) / (siny -2y)

tan line to the point (-1,1) on the curve x³-3x = y³+y

y'=(3x²-3)/(3y²+1) answer : 0

At what points does the curve y²=x³-3x have horizontal tangents

y'=0 x=±1 answer: (-1, √2) and (-1, -√2

tan line to the point (2,1) on the curve x²+2y²=6

y(x)=√(3-¹/₂x²) y'(x)= answer: y-1 = -1(x-2)

What is the linerization of the curve y=x² at (1,1)?

y= 2x-1

Product rule

ƒ(x)g(x)' + ƒ(x)'g(x)

A(circle)

πr²

V(sphere)

⁴/₃ πr³

Estimate sin46°

≈0.7191

Estimate ³√28

≈3.037

After 1 hour 10 grams of corundium decays into 6 grams of corundium and 4 grams of corinium. When will there be 1 gram of corundium?

≈4.5 hours