Calc I - Exam #3

9/2(x^2+1)

9tan-1(x-sqrt1-x^2)^2

0,2/9; exponents over each other

How do you find critical points of equation like t^2e^9t?

e^1/5

How do you find critical points of equation like x^-5 ln(x)?

-8; the second exponents numerator

How do you find critical points of equation like x^1/9 - x^-8/9?

0,8,16/7 (8 times 2 over exponent plus denominator)

How do you find critical points of equation like x^4/5(x-8)^2?

0,5/8; numerator first exponent and denominator second

How do you find critical points of equation like x^5e^-8x?

-Find derivative -Solve for 0 -Find value at all points (given and ones that equal 0)

How to find max/min values?

-f'c=f(8)-f(3)/8-3 -f(8)-f(3)/5≥4 -f(8)-f(3)≥20 then... -f(8)-f(3)/5≤5 -f(8)-f(3)≤25 20 and 25

If 4≤f'(x)≤5 for all values of x, what max and min values of f(8)-f(3)?

f'(c)=f(b)-f(a)/b-a -f'c=f(9)-f(4)/9-4 -f(9)-10/≥_3 -f(9)-10≥15 -f(9)≥25

If f(4)=10 and f'(x) ≥ 3 for 4≤x≤9, how small can f(9) be?

1

MVT of 2x^2-5x+1

-cont and diff on all real numbers so cont on [ ] and diff on ( ) --1/5ln(1/5(1-1/e^5))

MVT of e^-5x

-cont and diff on interval -c=(x-1)/lnx

MVT of ln(x)

-cont and diff on interval -9 = 3 sqrt 30 ; 7 = 2 sqrt 32 ; 6 = sqrt of 126; 5 = 3 sqrt 10; 2 = 3 sqrt 2

MVT of x/x+6

-cont on [ and diff on ( ) bc polynomials are cont and diff on all real #s -high chance = 2sqrt3/3

MVT of x^3 + x -3

[0,X]; answer is equal to x over 4!

Rolles Theroem

- Quadratic formula - FOIL of derivative (ie x^3+6x^2-36x) - look at top and bottom numbers of derivative (y-1/y^2-3y+3)

Ways to find critical points...

min = 0 max = 4(5)^3/4/25 (#-1)#^3/4 over number squared

absolute min and max of e^-x - e^-5x [0,1]

min = 1-ln(8) max = 2-ln(16)

absolute min and max of x - ln 8x. [1/2,2]

f(x) = e^x/sqrt(1-e^2x) domain f(x) = (-infinity, 0] domain f'(x) = (-infinity, 0)

arcsin(e^x)

8e^8x/1+e^16x

arctan(e8x)

1-x^2/1+x^2

cos(2tan^-1x)

(sin(2x))^x(2xcot(2x)+ln(sin2x)) reg. (2xcot2x plus ln reg)

derivative; (sin2x)^x

-lnx/sqrt(1-x^2) + arccosx/x

derivative; arccosx(lnx)

5/2sqrt(5x)sqrt(1-5x)

derivative; arcsin(sqrt5x)

10tan^-1(5x)/1+25x^2 (number with x times 2)tan^-1(#x) over 1 + (#^2)x^2

derivative; tan^-1(5x)^2

4x^4cosx (cosx/x-lnxsinx)

derivative; x^4cosx

e^8tsin(2t) (16tcos(2t)+8sin(2t)

diff; e^8tsin(2t)

32/(e^8u+e^-8u)^2

diff; e^8u - e-8u / e^8u + e^-8u

6/3y+1 - y/y^2+1 2 times # by y over numerator - y over inside of sqrt

differentiate; (3y+1)^2/sqrt(y^2+1)

6ln(5x) front # , ln, inside x

differentiate; 6xln(5x)-6x

4e^x(ln(1+e^x))^3/1+e^x

differentiate; [ln(1+e^x)]^4

e^tanpit * pi sec2(pit) dt reg * pi derv. tan (pi t ) dt

differentiate; e^tanpit

t/2+t^2 dt

differentiate; ln sqrt (2+t^2)

t/2+t^2 dt t over inside times dt

differentiate; ln sqrt (2+t^2)

-1-10x/4-x-5x^2 deriv of top over reg.

differentiate; ln | 4 - x - 5x^2 |

7x^6/(x^7+4) deriv of inside over reg

differentiate; ln(x^7+4)

5x^4/(x^5+6)ln10 deriv of inside over reg times ln10

differentiate; log10(x^5+6)

5cos(5ln(x))/x deriv of sin. & reg inside over x

differentiate; sin(5(ln(x))

1/sqrt(1-2x^2) * 2

differentiate; sin^-1(2x)

cos(x)ln(2x)+sinx/x deriv. of sin & reg inside + sinx/x

differentiate; sinxln(2x)

1/2zsqrt(8+lnz) dz 1 over 2z (reg) dz

differentiate; sqrt (8+ln z)

10e^x/sqrt 1+4e^5x

differentiate; sqrt 1+4e^5x

2x(4xcos(8x)+sin(8x))dx

differentiate; x^2 sin 8x

8x^2cos(8x) + 2xsin(8x) dx inside^2 (deriv of middle)(inside) plus (deriv of left)(middle)(inside) dx

differentiate; x^2sin8x

x^8x (8ln(x)+8) reg (#lnx+#)

differentiate; x^8x

ln(4+e^y) + ye^y/4+e^y ln (inside) plus ye^y over inside

differentiate; yln(4+e^y)

-c=DNE -this does not contract the MVT since f is not cont at x=3

f(x)=(x-3)^2, find values of c such that f(5)-f(2)=f'c(5-2)

-derivative; f'x=1/x^2+7x+15 * (2x+7) -f'x = 0 ; 2x+7 = 0 ; x = -3.5 -eval at -4, -3.5, and -1 (fill in)

find max and min of ln(x^2+7x+15) on [-4,1]

1/xln(6)+log6(e)

log6(xex)

2x/(x^2-25)ln(8)

log8(x^2-25)

1/sqrt4ex-1

sec-1(2ex)

x/sqrt(x^2+1)

sin(tan-1x)

6/srt(6x+1)^2

sin-1(6x+1)

4cotx+6sec2x/tanx-4/s^2+4

sin4xtan6x/(x^2+4)2

x/sqrt(1-x^2)

tan(sin-1x)

12tan^-1(6x)/1+36x^2

tan-1(6x)^2

8x^8cosx(cosx/x-lnxsinsx)

x^8cosx

-20v/(5+v^2)^2 dv

y = 5-v^2/5+v^2

7/(7+2s)^2 ds

y = s/(7+2s)

-usinu + cos u du

y = u cos u

sec2(sqrt(7t)) * 7/2sqrt(7t) dt

y=tan sqrt (7t)