Before the midterm

Dropping a term of the expression

The habit: Substitute into every part and carry each piece through to the end, constants included.

64 practice questions are built to catch it, in 17 topics.

Practise against it: 6 questions

What it looks like

A question from the definition of the derivative, with the answer this slip gives beside the right one.

Standard3 marks · 4 min

Use the limit definition of the derivative to find $f'(3)$ for $f(x) = x^{2} - 4x$ .

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

Rules such as the power rule earn no marks on this question.

With the slip: $\displaystyle 6$
Worked through: $\displaystyle f'(3) = 2$

−4(3 + h) is −12 − 4h. With only −12 the −4h is lost, and after dividing by h that is a −4 missing from the quotient.

See the worked solution

Where it comes up

Power functions and first sketches1 question
Functions, graphs and inverses2 questions
Limits6 questions
Continuity4 questions
The definition of the derivative7 questions
Differentiation rules4 questions
Exponential and trigonometric functions5 questions
The chain rule and logarithms1 question
Curve sketching4 questions
Optimization5 questions
Linear approximation1 question
Taylor polynomials3 questions
Newton’s method7 questions
Slope fields and Euler’s method1 question
Logistic growth and phase lines5 questions
Functions of two variables and partial derivatives5 questions
Optimization in two variables and Lagrange multipliers3 questions

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

01
A person’s core body temperature, in degrees Celsius, is modelled by the function below, where $t$ is the time in hours after midnight. At what time $t$ , with $0 \leq t < 24$ , is the temperature highest? $\displaystyle T(t) = 36.9 + 0.4\sin\!\left(\frac{\pi}{12}(t - 11)\right)$
Power functions and first sketchesStandard
02
Let $h(x) = \sqrt{1 + e^{2x}}$ . Suppose $h(x) = f\bigl(g(x)\bigr)$ with the inner function $g(x) = e^{2x}$ . Which formula gives the outer function $f$ ?
Functions, graphs and inversesStandard
03
Let $f(x) = 3e^{2x} - 1$ . Find $f^{-1}(5)$ .
Functions, graphs and inversesStandard
04
Evaluate the limit. $\displaystyle \lim_{x \to \infty} \Bigl(\sqrt{9x^{2} + 4x} - 3x\Bigr)$
LimitsStandard
05
Find the value of the constant $a$ for which the limit below holds. $\displaystyle \lim_{x \to 0} \frac{\sqrt{ax + 25} - 5}{x} = \frac{3}{2}$
LimitsStandard
06
Which statement about the vertical asymptotes of the graph of the function below is correct? $\displaystyle g(x) = \frac{x^{2} - 4}{\sqrt{x + 7} - 3}$
LimitsChallenging