Math 151A week 10

Review

100

csc(3π/2)

-1

200

Lim_x->∞ (11+8x)/(x³+7x)

0

300

h(t)=t³-t²cos(t)

h'(t)=3t²-2tcos(t)+t²sin(t)

400

f(x)=((2x)/(x²+1))³

What is f'(x)

f'(x)=3((2x)/(x²+1))²*((-2x²+2)/(x²+1)²)

500

f(w)=tan(w)sec(w)

What is f'(x)

f'(w)=sec3(w)+tan2(w)sec(w)

Implicit differentiation

100

Find y'

2y³+4x²-y=x⁶

y'=(6x⁵-8x)/(6y²-1)

200

Find y'

7y²+sin(3x)=12-y⁴

y'=(-3cos(3x))/(14y+4y³)

300

Find y'

4x²y⁷-2x=x⁵+4y³

y'=(-8xy⁷+5x⁴+2)/(28x²y⁶-12y²)

400

Find y'

e^x-sin(y)=x

y'=(1-e^x)/(-cos(y))

500

Daily Double

Find y'

cos(x²+2y)+xe^{y^2}=1

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

exponential functions

100

Log_3/8(27/512)=

3

200

How has f(x) shifted from the parent function y=2^x

f(x)=3^1+x

left 1 unit

vertical stretch by factor of 3/2

300

How has h(x) shifted from the parent function y=2^x

h(x)=2^3-x-7

left 3 units

rotate about the y-axis

vertical shift down 7 units

400

How has h(t) shifted from the parent function y=e^t

h(t)=8+3e^t-4

vertical shift 8 units up

vertical stretch by a factor of 3

horizontal shift 4 units right

500

How has g(z) shifted from the parent function y=e^z

g(z)=10-(1/4)e^-2-z

vertical shift up 10 units

vertically shrink by factor of 4

rotate about the x-axis

rotate about the y-axis

horizontal shift 2 units right

Inverse functions & Logs

100

expand

Log₇(10a⁷b³c^-8)

Log₇(10)+7Log₇(a)+3Log₇(b)-8Log₇(c)

200

expand

Log[z²(x²+4)³]

2Log(z)+3Log(x²+4)

300

expand

Ln[(w²t^3/4)/(t+w)^1/2]

2Ln(w)+(3/4)Ln(t)-(1/2)Ln(t+w)

400

combine into one logarithm

7Ln(t)-6Ln(s)+5Ln(w)

Ln(t⁷s^-6w⁵)

500

combine into one logarithm

2Log₃(x+y)+6Log₃(x)-(1/3)

Log₃[(x+y)²x⁶*3^-1/3]

Derivatives of Logs and exponents

100

f(x)=9Log₄(x)+12Log₁₁(x)

What f'(x)

f'(x)=9/(x*Ln(4)) +12/(x*Ln(11))

200

h(t)=6^t-4e^t

What is h'(t)

h'(t)=6^tLn(t) - 4e^t

300

R(t)=(t²-6t+3)*e^t

What is R'(t)

R'(t)=(2t-6)e^t + (t²-6t+3)e^t

400

Daily Double (Max bet: 1000)

Find the derivatives of the four functions

1) f(x)=a^x

2) g(x)=Log_a(x)

3) h(x)=Ln(x)

4)s(x)=e^x

500

g(r)=[r²+Log₇(r)]/[7^r]

what is g'(r)

g'(r)=[7^r(2r+(rLn(7))^-1-(7^rln(7))(r²+Log₇(r))]/[7^2r]