Math 150 Pre-Exam Review

Derivative Definition

100

1. f(x)= 6

f'(x)=0

200

2. V(t)=3-14t

v'(t)=-14

300

3. g(x)= x^2

g'(x)=2x

400

4. Q(t)=10+5t-t^2

Q'(t)=5-2t

500

5. W(z)=4z^2-9z

W'(z)=8z-9

Derivative Rules

100

1. ƒ (t) = (4t² − t) (t³ — 8t² + 12)

f'(t)=20t^4-132t^3+24t^2+96t-12

200

2. y=(1+ √x³) (x^-3 − 2 ³√x) 

y=1-2x^3³√x+√x-2x^3 6√x11/(x^3)

300

4. g(x)= 6x2/2-x

g'(x)=24x-6x^2/(2-x)^2

400

5. R(w)= 3w+w^4/ 2w2+1

4w^5+4w^3-6w^2+3/ (2w^2+1)^2

Inverse Trig/Chain Rule

100

1. T (z) = 2 cos(z) + 6cos-¹ (z)

T′(z)=−2sin(z)−   6/√1−z2

200

2. g (t) = csc-1 (t) - 4cot-1 (t)

g′(t)=−1/|t|√t2−1  +  4/t^2+1

300

3. y = sqrt 3(1-8z)

−8/3(1−8z)^-2/3

400

4. R(w) = csc(7w)

R′(w)=−7csc(7w)cot(7w)

500

5. G (x) = 2 sin(3x + tan(x))

G′(x)=2(3+sec2(x))cos(3x+tan(x))

Implicit

100

6. h (u) = tan(4+10u)

h′(u)=10sec2(4+10u)

200

6. ex – sin(y) = x 

y′=(1−e^x)/−cos(y)=

(e^x−1)sec(y)

300

7. 4x²y7 — 2x = x5 + 4y³

y′=(8xy^7−5x^4−2) / (12y^2−28x^2y^6)

Graph/Log

100

5. f(x)=2x3-9x2-60x

x=−2,x=5 

Increasing : (−∞,−2)&(5,∞)Decreasing : (−2,5)

x=−2: Relative Maximumx=5: Relative Minimum

200

6. h(t)=50+40t3 - 5t4-4t5

take 1st derivative

t=−3,t=0,t=2

Increasing : (−3,0)&(0,2)Decreasing : (−∞,−3)&(2,∞)

t=−3: Relative Minimum

 t=0:Neither 

t=2: Relative Maximum

300

7. y=2x³- 10x²+12x-12

Increasing : (−∞,5−√7/3)&(5+√7/3,∞)Decreasing : (5−√7/3,5+√7/3)

x=5−√7/3=0.78475: Relative Maximumx=5+√7/3=2.54858: Relative Minimum

400

f(x)=(5−3x^2)^7√6x^2+8x−12

=7ln(5−3x^2)+12ln(6x2+8x−12)

500

2.y=sin(3z+z2)/(6−z4)3 

sin(3z+z2)/(6−z4)^3[(3+2z)cot(3z+z2)+ (12z^3/6−z^4)]