Calculus

Differentiation

100

f(x) = e^(-3x). Find f''(x)

9e^(-3x)

200

f'(x) = x^3 - 9x^2 +24x +3. Write the down the x-coordinates of the two points of inflection.

x=2, 4

300

A particle moves with velocity v(t)=2t - 0.3t^3 +2, for t is greater than 0

Find the acceleration of the particle after 2.2 seconds.

-2.36

400

Consider f(x)=x^3 - (p/x) ,  x≠0

There is a minimum value of f(x) when x=1. Find the value of p.

-3

500

Consider the function f(x) = 2x/cosx. Find f'(π )

-2

Integral Calculus

100

∫(5,1) f(x)dx = 6. Find ∫(5,1) 2f(x)dx

12 

200

f'(x) = 3x^2 - 3. Given that f(2) = 1, find f(x)

f(x)= x^3 - 3x - 1

300

v(t)=3e^(2t) + 2t. Find an expression for the displacement of the particle.

3e^(2t)/2 + t^2 +C

400

f(x) = ∫ 8/(2x-1)dx, for x is less than 1/2. The graph of f passes through(1,5). Find f(x)

f(x)= 4ln(2x-1)+5

500

∫(k,0) dx/(2x+1) = 1, Find the value of k

k= (e^2 - 1)/2

Warm-up

100

f=ln(3x), find the first and second derivative

f'(x) = 1/x

f''(x) = -1/(x^2)

200

f(x)=(2x-5)^3. Find f'(x)

6(2x-5)^2

300

integrate 1/x²

-1/x+C

400

integrate 

∫(1,2) 5−x² dx

12

500

The graph of f passes through point(π /12, 2). Given that f'(x)=2cos(2x), find f(x)

 

f(x)=sin2x+3/2

misc(hard?)

100

Paper 2: 

f(x)=e^2sin(πx/2), for x> 0. 

The kth maximum point on the graph of f has x-coordinate x_k where k=all real values. Given that x_k+1=x_k+a, find a

a=4

200

Consider f(x), g(x), and h(x), for XER where h(x)=f(g(x)

Given that g(3)=7, g'(3)=4, and f'(7)=-5, find the gradient of the normal to the curve of h at x=3

gradient of normal = 1/20