Exam 2 Review

Position functions and tangent lines

100

Using s(t)=t³+6t+8, what is the position of the particle at 2 seconds?

s(t)=28 inches

200

Using s(t)=t³+6t+8, what is the average velocity of the particle from [ 1 , 3 ] ?

Average velocity: 21 in/s

300

Using s(t)=t³+6t+8, what is the acceleration of the particle at 2 seconds?

12 in/s²

400

Using s(t)=t³+6t+8, find the acceleration of the particle when its velocity is 33 in/s.

18 in/s²

500

Find Ltan:f(x)=4x³+e^x-sinx; x=3

round to four decimal places

Ltan: y=129.0755x-1.1311

Name that derivative!

100

y=lnx, y'=

y'= 1/x (sometimes times k)

200

y=sinx, y'=

g(x)=cosx, g(x)'=

y'=cosx

g(x)'=-sinx

300

y=secx, y'=

g(x)=tanx, g(x)'=

y'=secx x tanx

g(x)'=sec²x

400

y=cscx, y'=

g(x)=cotx, g(x)'=

y'=-cscx x cotx

g(x)'=-csc²x

500

y=log_ax, y'=

y'=1/x(ln_a)

Product/Quotient Rule

100

y=(2x/4x²)

y'=[-8x²/(4x²)²]

quotient rule

200

f(x)=(3x²-1)/(2x+5)

f(x)=[2(3x²+15x+1)/(2x+5)²]

quotient rule

300

y=lnxcosx

y'=(cosx/x)-lnxsinx

product rule and trig/log functions

400

f(x)=square root of x(1-x2)

f(x)'=[(-5x²-1)/(2 times the square root of x)

500

g(x)=2^x(3cosx)

g(x)'=(2^x x ln₂ x 3cosx)+(-3sinx x 2^x)

product rule, a^x, and trig/log functions

Differentiation Rules

100

y=(x²+6x-3)+(x³-4)    

y'=(2x+6)+(3x²)

power rule + addition rule

200

y=(1/4)e^x+x⁴-2

y'=(1/4)e^x+4x³

power rule and ex

300

y=2^x

y'=2^x x ln2

y=a^x -> a^x x lna

400

f(x)=16[(x³-2x²)-(4x⁴+6x⁵-3)]

f(x)'=16[(3x²-4x)-(16x³+30x⁴)]

power rule, subtraction rule and constant rule

500

y=x^e

y'=ex^e-1

the 'e' trick

Chain Rule

100

y=(6x²-5)⁴, y'=

y'=48x(6x²-5)³

200

y=(1/3x³+4)^-1, y'=

y'=[-x/(1/3x³+4)²]

300

y=3ln³x, y'=    

y'=[9(3lnx)³/x]

400

y=e^(x^^2+3x), y'=

y'=e^{(x^2+3x)} x (2x+3)

500

y=csc(2 root x), y'=

y'=-csc(2 root x) x cot(2 root x) x x^-1/2