Unit 2 Review - Differentiation,Trig & Log Functions

Rates of Change

100

Differentiate:

f(x) = 0.5x^3 +5/x -7

f'(x) = 1.5x^2 -5/x^2

200

Differentiate, leaving your answer with a positive index:

p(x) = 1/sqrtx

p'(x) = -1/(2sqrtx^3)

300

Find f'(x) when:

f(x) = (1-4x)^5

 

f'(x) = -20(1-4x)^4

400

Differentiate, leaving your answer with a positive index:

m(x) = 5/(x+2)

 

m'(x) = -5/((x+2)^2

500

Determine the x-coordinates of the stationary points for:

y = x^3 +6x^2 -15x+2

x = -5 and x = 1

Derivatives

100

f(x)= x(x+1)³

Find f'(x) (fully factorise me!)

f'(x)=(x+1)²(4x+1)

200

What do you have if f'(x)=0 and f''(x)<0

Maximum Stationary Point

300

Tech Free     If 

f(x)=x^3/10 - x^2

State whether f(x) is increasing or decreasing at x = 2

f'(x) = (3x^2)/10 - 2x

f'(2) = 12/10 - 4

f'(2) = -2.8 Negative number = Decreasing

400

Calculate the gradient of the tangent to the curve at x = 4

f(x) = sqrt(x)

m = 1/4

500

Use the second derivative to determine the stationary point and its nature of:

f(x) = 3x^2 - 2x

Stationary point (1/3, -1/3) Minimum as 

f''(x) = 6>0

Log Functions

100

Simplify:

1/2 log₃16 + 2log₃4

log₃64

200

Solve:

log₂(x - 3) = 5

x = 35

300

State the equation of the asymptote and the x and y intercepts of:

y = 2^x -4

Asymptote  y = -4

Y - intercept  y = -3

X - intercept  x = 2

400

State the domain and range of:

y = log₂(x + 2)

Domain: x > -2

Range:  R

500

Tech Active    Solve to the nearest whole number:

560 = 135 * 1.7^0.02t + 20

t = 131

Try your luck!

100

  Find the average rate of change of:

W(t) = t² - t +1  (W = weight in grams)

between 1 and 3 seconds.

3 g/s

200

Graph the gradient function of y=x²

y = 2x is graphed (A diagonal line)

300

Tech Active: Solve to 3 decimal places:

 

-100k = log_2.7(0.5)

k = -0.002

400

Tech Free: A probability function is defined as: 

p(x) = 1/9 (4 - x)   x{0, 1, 2}

Construct a probability distribution table and find E(X).

E(X) = 7/9 
 

500

A rectangular fish tank has a square base with its height being equal to half of its base. 

Write an expression for its volume and then find the rate of change when h = 1m.

V = 4h³

V'(1) = 12m³/m

Tech Active

100

The temperature (in degrees Celsius) is given by 

T(h) = h^2 -4h + 22

-sqrt3/2

200

Given  cos theta = 5/6   write the exact value of 

sin theta

sqrt11/6

300

State the am

y= (2x-5)^(1/2)

Amplitude = 2

Period = 

6pi

400

 Solve:

sqrt2 cos(x) + 1 = 0 

 {0lexle2pi} 

x = (3pi)/4, (5pi)/4

500

Tech Active: An arc subtends an angle of  56 degrees at the centre of a circle of radius 10cm.

Calculate the length of the arc.

 

9.77cm