Calculus and Vectors Grade 12

Limits

100

Evaluate if the limit exists

Lim 5

x->-1

=5

200

Evaluate if this limit exists 

Lim x²-4/x²⁺³

^x->-1

(-1)²-4/(-1)²+3

=-3/4

300

Evaluate if the limit exists

Lim   (x-3)(5x²+2)

x->0

=(0-3)(5(0²)+2

=(-3)(2)

=-6

400

Evaluate if the limit exists

Lim       x²-2x-3/x-3

x->3

=(x-3)(x-1)/(x-3)

=x+1

=3+1

=4

500

Evaluate if the limit exists

Lim        (1+x)²-4/x+3

x->-3

=1+2x+x²-4/x+3

=x²+2x-3/x+3

=(x-1)(x+3)/(x+3)

=x-1

=-3-1

=-4

Derivatives

100

Differentiate the following

3x⁵

y=3(5x⁴)

y¹(x)=15x⁴

200

Differentiate the following

1/x

=x^-1

=-x^-2

=-1/x²

300

Use the Product Rule to differentiate

y=(2x-3)(1-x)

y=(2x-3)(-1)+(1-x)(2)

y=-2x+3+2-2x

y=-4x+5

400

Find the second derivative of the following

1/x

f(x)=x^-1
f^l(x)=-x^-2
f^ll(x)=2x^-3​

=2/x³

​​​

500

Find the equation of the tangent line to the curve 

f(x)=(3x²+2)(2x³-1) at x=1

y=36x-31

Vectors

100

Write this true bearing as a quadrant bearing 

035^o

N35⁰E

200

A Parallelogram is labelled ABCD, state all the pairs of equivalent vectors.

AB=DC, DA=CB, BA=CB,AD=BC

300

Simplify the following vector

2u+v+3u-v

5u

400

Simplify the Following Vector

3(u+v)+2u

=3u+3v+2u

=5u+3v

500

Draw the magnitude and direction of the following resultant vector.

The velocity of water is given using the rectangular vector components of 2.5 m/s N45^oW and 3.5 m/s S45^oN

x²=3.5²=2.5²                  tan0=3.5/2.5

x²=18.5                        Theta= 54.5^o

x=4.3 m/s

Lines and Planes

100

Write a vecor equation for the line through the points A(1,4) and B(3,1)

(x,y)=(1,4)+t(2,-3)

200

Find the coordinates of 2 points on the line

x=3+2t and y=-5+4t

t=1                            t=0 

x=3+2(1).                  x=3+2(0)

x=5.                           x=3

y=-5+4(1).                 y=-5+4(0)

y=-1.                          y=-5

(5,-1)                          (3,-5)

300

Write the vector and parametric equations for the line through points A(2,-1,5) and B(3,6,-4)

Vector:(x,y,z)=(2,-1,5)+t(1,7,9)

Parametric: 

x=2+t

y=-1+7t

z=5+9t

400

Consider the plane defined by the scalar equation x+2y-z-8=0.

Determine If the point 1,6,5 is on the plane

0=1+2(6)-5-8

0=1+12-5-8

0=0

This point is on the plane

500

Determine the number of solutions and how the planes intersect, solve if possible.

x+3y-z=-10

2x+y+x=8

x-2y+2z=-4

1) y+3y-z=-10.                    2)2x+y+z=8

3)x-2y+2z=-4.                    2x3)2x-4y+8z=-8

4)5y+3z=-6.                        5)5y-3z=16

4)5y+3z=-6

5)5y-3z=16

0=22

Not parallel, the planes intersect in pairs, no solution

Rates of change and Quotients

100

What is the formula to calculate the average rate of change as a difference quotient? 

f(a+h)-f(a)

------------

     h

200

Write a difference quotent that can be used to estimate the instantaneous rate of change of y=3x²-5x at x=-1

Dq=3h-11

300

Determine an expression that represents the average rate of change of the height above the ground in terms of a and h

h(t)=90-4.9t

=-9.8a-4.9h

400

Determine the average rate of change in the interval from 1-3 seconds.

Equation: -9.8a-4.9h

a=1

h=2

=-9.8(1)-4.9(2)

=-19.6

500

An oil tank is being drained, The volume of oil remaining in the tank is represented by the function V(t)=60(25-t)²

Estimate the instantaneous rate of change of the volume at each interval:5,10,15,20.

5=-2400L/m

10=-1800 L/m

15=-1200 L/m

20=-600 L/m