About Engineering Mathematics
Engineering mathematics at this level means the calculus, matrices and series that every engineering paper opens with: differentiating and integrating the standard expressions, taking limits that look like 0/0, evaluating determinants, and adding up the terms of a series. Nearly every question is settled by working through it in a fixed order, which is what makes the marks easy to hold on to.
What you need to understand
- Differentiate term by term: the derivative of x^n is n x^(n-1), and the derivative of a constant is zero.
- The second derivative is the derivative of the derivative. It is what tells you whether a stationary point is a maximum or a minimum.
- Integrating undoes differentiating: raise the power by one and divide by the new power. For a definite integral, put in the upper limit, put in the lower limit, and subtract.
- A limit of the form 0/0 is not an answer. It means the top and the bottom share a factor that has to be cancelled before the value can be found.
- In a limit x tends to a value, the expression never actually reaches that value, so cancelling the shared factor is legitimate.
- A determinant of a 3x3 matrix is found by expanding along any row: each entry multiplies the 2x2 determinant left when its row and column are struck out, and alternate terms take a minus sign.
- A series is arithmetic if the difference between consecutive terms is constant, and geometric if the ratio is constant. Arithmetic series have a common difference; geometric ones have a common ratio.
- The sum of the first n terms of an arithmetic series is n/2 times the sum of the first and last; for a geometric series it is the first term times (r^n - 1)/(r - 1) when r is not 1.
How to work through these questions
- For a derivative, write the function out and differentiate one term at a time rather than trying to do it in your head. Check the sign of every term as you go.
- For a definite integral, find the indefinite integral first, then put in the two limits. A negative lower limit is where most marks are lost - square it as a positive number.
- For a limit that gives 0/0, factorise the top and the bottom and cancel the shared bracket. What is left can be evaluated at the point.
- For a determinant, pick the row or column with the most zeros. Each zero saves you the work of one 2x2 determinant.
- For a series, write down the difference or the ratio between the first two terms and check it against the next pair. Only then decide which kind of series it is.
- For a sum of terms, remember the question asks for the sum and not the last term - find the last term first, and use it in the sum, rather than giving it as the answer.
Mistakes that cost marks
- Differentiating when the question says integrate, or the other way round. The power goes up when integrating and down when differentiating.
- Forgetting to divide by the new power when integrating. The commonest single mark lost in this topic.
- Dropping the minus sign when the lower limit of a definite integral is negative, or subtracting the limits the wrong way round.
- Treating 0/0 as the answer to a limit instead of as a signal to factorise and cancel.
- Expanding a determinant along a row without the alternating minus sign on the second term.
- Giving the last term of a series as the sum, or using the wrong number of terms - for example adding 11 terms when the question asks for the first 10.
- Reading a geometric series as arithmetic because the first two terms happen to fit both, without checking the third.
Worked example
What is the value of the definite integral of 3x^2 + 2x from x = 1 to x = 3?
- Integrate term by term. The integral of 3x^2 is x^3, and the integral of 2x is x^2.
- The indefinite integral is therefore x^3 + x^2 - that is, raise each power by one and divide by the new power.
- Put in the upper limit x = 3: 27 + 9 = 36.
- Put in the lower limit x = 1: 1 + 1 = 2.
- Subtract the lower value from the upper one: 36 - 2 = 34.
- The value of the integral is 34. The same answer comes out of Simpson rule over a thousand strips, which is how it is checked.
Answer: 34
Practice questions with answers
A few Engineering Mathematics questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Work out the derivative of f(x) = -3x^2 + 4x - 2.
-
A
-6
-
B
-6x + 2
-
C
-3x^2 + 4x - 2
-
D
-6x + 4
Answer: Option D — with explanation
Differentiate f(x) = -3x^2 + 4x - 2 term by term: the derivative of a constant is 0, and the derivative of x^n is n x^(n-1).
That gives -6x + 4. The function itself, its second derivative, and the same derivative with one coefficient changed are all different functions, and none of them is the answer.
Common mistakes
- -6 is not the answer: -6
- -6x + 2 is not the answer: -6x + 2
- -3x^2 + 4x - 2 is not the answer: -3x^2 + 4x - 2
Question 2
Find the value of the definite integral of 4x^2 - 3x - 4 from x = -3 to x = 4.
-
A
136/3
-
B
497/6
-
C
-497/6
-
D
2387/6
Answer: Option B — with explanation
Integrate 4x^2 - 3x - 4 term by term, raising each power by one and dividing by the new power, then put in the two limits and subtract: the value at x = 4 minus the value at x = -3.
That gives 497/6. Forgetting the lower limit, dropping a minus sign, or raising the power without dividing by it all give something else.
Common mistakes
- 2387/6 is not the answer: 2387/6
- -497/6 is not the answer: -497/6
- 136/3 is not the answer: 136/3
Question 3
Work out lim x→-2 (-x^2 + 4x + 12) / (-x - 2).
Answer: Option C — with explanation
Putting x = -2 into (-x^2 + 4x + 12) / (-x - 2) gives 0/0, which decides nothing yet - both parts carry the factor (x - -2).
Cancelling that factor leaves (x - 6), and the limit is what that gives when x is replaced by -2, which is -8. The number 6 is where the cancelled expression is zero, and -2 is where it is not defined at all; neither of those is the limit.
Common mistakes
- 4 is not the answer: 4
- -2 is not the answer: -2
- 6 is not the answer: 6
Question 4
What is the determinant of the matrix
[ 1 -3 1 ]
[ -2 5 -5 ]
[ 1 6 0 ]?
Answer: Option B — with explanation
Expand along the first row: each entry in that row is multiplied by the two-by-two determinant left behind when its own row and column are struck out, and the middle one takes a minus sign.
For this matrix that gives 28. The trace, the product of the leading diagonal, and the sum of all nine entries are all different numbers.
Common mistakes
- 0 is not the answer: 0
- -28 is not the answer: -28
- 6 is not the answer: 6
Question 5
Find the sum of the first 8 terms of the series -4, -12, -36, -108, ...
-
A
-13120
-
B
-39364
-
C
-8748
-
D
-4372
Answer: Option A — with explanation
The series is geometric: each term is 3 times the one before. With 8 terms, written out, it runs -4, -12, -36, ... up to the 8th term, which is -8748.
Adding those 8 terms gives -13120. The last term on its own, and the sum of a different number of terms, are the two answers that come up when the count or the last term is used in place of the sum.
Common mistakes
- -39364 is not the answer: -39364
- -8748 is not the answer: -8748
- -4372 is not the answer: -4372
Frequently asked questions
Do I always have to factorise for a limit?
Only when putting the value in gives 0/0 or something else undefined. If the expression can be evaluated straight away, that value is the limit and there is nothing to cancel.
Which row should I expand a determinant along?
Whichever row or column has the most zeros, because each zero removes one of the 2x2 determinants you have to work out. Any row or column gives the same answer.
How do I tell an arithmetic series from a geometric one?
Subtract the first term from the second, and the second from the third. If both differences are the same it is arithmetic. If both ratios are the same, it is geometric. Always check the third term - the first two fit either.
What if my answer is not one of the options?
Check the sign first, then the arithmetic on the limits or the power. Nearly every wrong option offered in these questions is there because of one of those two, so comparing your answer with the options often shows which slip was made.
