# SECTION 3: CALCULUS I

## Learn the basics of Differentiation and Integration.

DIFFERENTIATION

At the end of this section, students should be able to:

• use the concept of the derivative at a point $x=c$ as the gradient of the tangent to the graph at $x=c$;
• use the $f'(x)$ and $\frac{dy}{dx}$ notation for the first derivative of $f(x)$;
• use $\frac{d}{dx}x^n=nx^{n-1}$ where $n$ is any real number;
• use $\frac{d}{dx}\sin x=\cos x$ and $\frac{d}{dx}\cos x=-\sin x$
• use simple rules of derivatives to find derivatives of sums and multiples of functions;
• calculate derivatives of polynomials and trigonometric functions;
• apply the chain rule in the differentiation of composite functions;
• differentiate products and quotients of simple polynomials and trigonometric functions;
• use the concept of the derivative as a rate of change;
• use the concept of stationary points;
• locate stationary points, maxima and minima, by considering sign changes of the derivative;
• calculate the second derivative, $f''(x)$;
• interpret the significance of the sign of the second derivatives;
• use the sign of the second derivative to determine the nature of stationary points;
• obtain equations of tangents and normal to curves.

INTEGRATION

At the end of this course, students should be able to

• recognise integration as the reverse process of differentiation;
• use the notation $\int f(x)$;
• show that the indefinite integral represents a family of functions which differ by constants;
• use simple integration rules;
• integrate functions of the form $(ax+b)^n$ where are real numbers and $n\neq -1$;
• find the indefinite integrals using formulae and integration theorems;
• integrate simple trigonometric functions;
• compute definite integrals;
• formulate the equation of a curve given its gradient function and a point on the curve;
• apply integration to:

1. find the area of the region in the first quadrant bounded by a curve and the lines parallel to the $y-$ axis;

2. find volumes of revolution about the $x-$ axis, for polynomials up to and including degree

Sean Hunte

Sean Hunte, Your Passionate Mathematics Teacher

I was once a student just like you. I struggled with Mathematics for the first two (2) years of my secondary school life. In third form my teacher, Dr Sargeant, taught Mathematics in a unique way which allowed me to successfully learn the subject which I was passionate about. Her teaching enabled me to thrive in an area which I would eventually pursue as a career.

Therefore, I have successfully completed CXC's CSEC General Mathematics and CAPE Pure Mathematics Units 1 and 2 before obtaining a degree in Economics and Mathematics.

I have ten (10) years of experience teaching Mathematics at the secondary level in Barbados. During this time I have taught hundreds of students who have successfully completed the following courses:

• CSEC GENERAL MATHEMATICS
• CAPE PURE MATHEMATICS UNIT 1
• CAPE PURE MATHEMATICS UNIT 2
• SAT MATHEMATICS

Your story will be different from mine but both of us desire the same result. SUCCESS!

I know that for many students Mathematics can be a struggle. I am therefore here to offer assistance in making the learning of Mathematics easier.

## Courses Included with Purchase

Learn how to Differentiate.
Sean Hunte
$10 INTEGRATION For Additional Mathematics Integration techniques for Additional Mathematics Sean Hunte$10