Course

O Level Additional Mathematics: Calculus (4037)

Name: O Level Additional Mathematics: Calculus (4037)
Price: 1299 BDT

Differentiation, its applications, integration, area and kinematics for Cambridge O Level Additional Mathematics 4037, taught step by step from the gradient of a curve.

71 lessons13h 39mEnglish

What you'll learn

Explain the derivative as the gradient of a curve and use the notation f′(x), dy/dx and d²y/dx²
Differentiate powers of x, e^x, ln x, sin x, cos x and tan x, with the chain, product and quotient rules
Find tangents, normals, small changes and connected rates of change
Find stationary points and decide maximum or minimum with the first or second derivative test
Solve practical maximum and minimum problems with boxes, cylinders and other shapes
Integrate standard functions, find the constant from a point, and evaluate definite integrals
Find areas under curves, below the x-axis, and between a line and a curve or two curves
Solve straight-line motion problems and draw and use displacement, distance, velocity, speed and acceleration graphs

Requirements

Cambridge O Level Mathematics 4024 or equivalent, including indices, surds and straight-line graphs
Radians, logarithms and e from the sibling Additional Mathematics courses, needed from chapter 3

Course content

71 lessons · 13h 39m

14.1-14.4 · Differentiation

The gradient function, the notation for derivatives, powers of x, the chain rule, the product and quotient rules, and second derivatives.

0/10

The gradient function

A curve has a different steepness at every point. The gradient function gives that steepness, the gradient of the tangent, for any x.

Preview12m 2.

Differentiating powers of x

Bring the power down and take one off it. Written f′(x) or dy/dx, and it works term by term on sums and multiples.

Preview12m

Negative and fractional powers

Rewrite roots and fractions as powers of x first. Then the same power rule works for any rational power.

12m

The chain rule

For a function inside a function, differentiate the outside, keep the inside, then multiply by the derivative of the inside.

14m

The product rule

When two functions of x are multiplied, the derivative is u times dv/dx plus v times du/dx.

13m

The quotient rule

When one function of x is divided by another, use v du/dx minus u dv/dx, all over v squared. The order matters.

13m

Second derivatives

Differentiate twice to get f′′(x) or d²y/dx². It tells you how fast the gradient itself is changing.

11m

Differentiation: practice

A timed drill of mixed questions: powers, the chain rule, products, quotients and second derivatives.

Differentiation: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

10.

Differentiation: summary

Every rule from this chapter, the common mistakes and the notation, on one screen.

14.5-14.9 · Applications of differentiation

Tangents and normals, small increments, connected rates of change, stationary points, the first and second derivative tests, and practical maximum and minimum problems.

0/13

Gradients and tangents

Put the x-value into dy/dx to get the gradient at a point. With the point, that gives the equation of the tangent.

Preview12m 2.

Normals

The normal is at right angles to the tangent. Its gradient is minus one over the tangent gradient.

Preview12m

Small increments and approximations

For a small change δx, the change in y is about dy/dx times δx. Use it to estimate a change without working the new value out in full.

12m

Connected rates of change

When y depends on x and x changes with time, dy/dt equals dy/dx times dx/dt.

13m

Connected rates of change: area and volume

Write the area or volume as a formula in one length, differentiate it, then link the rates. Put in the numbers only at the end.

14m

Stationary points

At a stationary point the gradient is zero. Solve dy/dx = 0, then find the y-value for each answer.

12m

The first derivative test

Check the sign of dy/dx just before and just after the point. Plus to minus is a maximum, minus to plus is a minimum.

12m

The second derivative test

Put the x-value into d²y/dx². Negative means a maximum, positive means a minimum, and zero means use the first test.

12m

Practical maximum and minimum problems

Write the quantity to make biggest or smallest in one variable, differentiate, solve, then test and answer in context.

15m

10.

Practical maximum and minimum problems: cylinders

Use the fixed volume to remove one variable, then find the radius that uses the least metal for a water tank.

15m

11.

Applications of differentiation: practice

A timed drill of mixed questions: tangents, normals, rates, small changes, stationary points and practical problems.

15m

12.

Applications of differentiation: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

13.

Applications of differentiation: summary

Tangents, normals, rates, small changes and the two tests for maximum and minimum, on one screen.

14.3-14.7, 14.9 · Differentiation of exponential, logarithmic and trigonometric functions

Derivatives of e to the x, ln x, sin x, cos x and tan x, with the chain, product and quotient rules, then tangents, rates and stationary points that use them.

0/10

Derivatives of exponential functions

The gradient of y = e^x is equal to its height. With the chain rule, e^(ax+b) differentiates to a e^(ax+b).

Preview12m 2.

Derivatives of logarithmic functions

The derivative of ln x is 1/x. For ln of a function, it is the derivative of the inside over the inside. Use the log laws first.

Preview13m

Derivatives of sine and cosine

With x in radians, sin x differentiates to cos x and cos x to minus sin x. Add the chain rule for sin(ax+b).

13m

The derivative of tan x

Write tan x as sin x over cos x and use the quotient rule to get sec² x. Then handle powers such as sin² x with the chain rule.

12m

Products and quotients with exponential, logarithmic and trigonometric functions

Pick the right rule for x²e^x, x ln x or sin x over x, and tidy the answer by taking out common factors.

13m

Further applications of differentiation: tangents and rates

Find tangents, normals and rates of change for curves that use e, ln and trigonometric functions, giving exact answers.

14m

Further applications of differentiation: stationary points

Solve dy/dx = 0 when it holds e, ln or sin and cos, then decide maximum or minimum with a full reason.

14m

Exponential, logarithmic and trigonometric derivatives: practice

A timed drill of mixed questions on e, ln, sin, cos and tan, with every rule from the chapter.

14m

Exponential, logarithmic and trigonometric derivatives: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

10.

Exponential, logarithmic and trigonometric derivatives: summary

The six standard derivatives, the rules that combine them and the common mistakes, on one screen.

14.10-14.12 · Integration

Integration as differentiation reversed, the constant of integration, powers of x, (ax+b) to a power, 1/x, exponential, sine, cosine and sec squared.

0/11

Differentiation reversed

If you know the gradient function, you can work back to the curve. There is a whole family of answers, one for each constant.

Preview12m 2.

Indefinite integrals

Add one to the power and divide by the new power, term by term, and always add + c.

Preview13m

Finding the constant of integration

Given dy/dx and one point on the curve, integrate, then put in the point to find c and the equation of the curve.

12m

Integration of functions of the form (ax + b)^n

Add one to the power, divide by the new power and also divide by a, the number in front of x.

13m

Integration of 1/x and 1/(ax + b)

The power rule fails for x to the minus one. Reverse the ln x derivative instead: the integral of 1/x is ln x + c.

12m

Integration of exponential functions

e^x integrates to itself. For e^(ax+b), divide by a.

12m

Integration of sine and cosine functions

Reverse the trigonometric derivatives: sin, cos and sec squared of (ax+b), dividing by a each time. Angles are in radians.

13m

Further indefinite integration

Differentiate a given function first, then use that result backwards to find a harder integral. The "hence" questions.

14m

Integration: practice

A timed drill of mixed indefinite integrals, with the constant found from a point in some of them.

14m

10.

Integration: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

11.

Integration: summary

Every standard integral, the constant of integration and the common mistakes, on one screen.

14.13 · Definite integration and area

Evaluate definite integrals, then use them for the area under a curve, areas below the x-axis, and areas between a line and a curve or two curves.

0/9

Definite integration

Integrate, put in the top limit, put in the bottom limit and subtract. The constant cancels, so leave it out.

Preview12m 2.

Further definite integration

Definite integrals of e, 1/x, sin and cos, giving exact answers with e, ln and π where the question asks.

Preview13m

Area under a curve

The area between a curve, the x-axis and two vertical lines is the definite integral between those x-values.

13m

Areas below the x-axis and sums of areas

Below the x-axis the integral comes out negative. Split at the crossing points and add the sizes of the parts.

13m

Area of a region bounded by a line and a curve

Find where the line meets the curve. Then take the area under the top one and subtract the area under the bottom one.

14m

Area between two curves

Integrate the top curve minus the bottom curve between the points where they meet.

14m

Definite integration and area: practice

A timed drill of mixed definite integrals and area questions.

14m

Definite integration and area: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

Definite integration and area: summary

Limits, signed areas, the split at the axis and top minus bottom, on one screen.

14.14-14.15 · Kinematics

Displacement, velocity and acceleration of a particle in a straight line, found by differentiating and integrating, and the five motion graphs.

0/12

Displacement and velocity

Displacement is position from a fixed point. Velocity is its rate of change, v = ds/dt, and its sign gives the direction.

Preview12m 2.

Acceleration

Acceleration is the rate of change of velocity, a = dv/dt. Differentiate twice from displacement.

Preview12m

Instantaneous rest and change of direction

A particle is at rest when v = 0. It changes direction only if v also changes sign there.

13m

Velocity from acceleration

Integrate acceleration to get velocity, then use the starting velocity to find the constant.

13m

Displacement from velocity

Integrate velocity to get displacement. Use the starting position to find the constant.

13m

Distance travelled

Distance is not displacement when the particle turns back. Split the time at each turning point and add the sizes.

14m

Displacement–time and distance–time graphs

Sketch s against t from its formula. The gradient is the velocity. The distance–time graph never goes down.

13m

Velocity–time and speed–time graphs

On a velocity–time graph the gradient is acceleration and the area is displacement. Speed is the size of velocity, so reflect the parts below the axis.

13m

Acceleration–time graphs

Draw a against t from its formula. The area under it is the change in velocity.

12m

10.

Kinematics: practice

A timed drill of mixed motion questions: differentiate, integrate, find rest times, distances and graph features.

14m

11.

Kinematics: exam practice

Past-paper style questions with full worked solutions and the marks each line earns.

15m

12.

Kinematics: summary

The chain s, v, a, which way to differentiate or integrate, the five graphs and the common mistakes, on one screen.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.