2. Differentiation: Definition and Fundamental Properties

Question 1

Graphing calculator required

During a two-hour filling test, the water depth in a tank is modelled by

$D(t)=t e^t+2$

for $0\leq t\leq2$ , where $t$ is measured in hours and $D(t)$ is measured in metres. The graph of $D$ and the line tangent to the graph at $t=1$ are shown. Calculator answers should be rounded to three decimal places.

Part (a)

[2]

Calculate the average rate of change of the water depth from $t=0.5$ to $t=1.5$ . Include units.

Part (b)(i)

[1]

Estimate $D'(1)$ by using two readable points on the tangent line shown in the graph.

Part (b)(ii)

[1]

Interpret the meaning of your estimate from part (b)(i) in the context of the tank.

Part (c)(i)

[1]

Calculate an exact expression for $D'(t)$ .

Part (c)(ii)

[1]

Represent the line tangent to the graph of $D$ at $t=1$ by an equation.

Part (d)(i)

[1]

Write an expression that can be solved to determine when the water depth is increasing at $6,\mathrm{m/hour}$ .

Part (d)(ii)

[1]

Determine the time at which this occurs.

Part (d)(iii)

[1]

Interpret your answer from part (d)(ii).

Question 2

Graphing calculator not permitted

The graph of a differentiable function $f$ contains the three labelled points shown. Using the closest points that are symmetric about $x=1$ , which value is the best estimate of $f'(1)$ ?

A. $1$

B. $2$

C. $4$

D. $8$

[1]

Select Answer

Question 3

Graphing calculator required

The concentration of a substance in a mixing tank is modelled by

$C(t)=\dfrac{t^2+1}{e^t+1}$

for $0\leq t\leq4$ , where $t$ is measured in hours and $C(t)$ is measured in $\mathrm{mg/L}$ . The graph of $C$ and its tangent line at $t=1$ are shown. Calculator answers should be rounded to three decimal places.

Part (a)(i)

[1]

Represent the average rate of change of $C$ from $t=0$ to $t=2$ by an expression.

Part (a)(ii)

[1]

Calculate this average rate of change and include units.

Part (b)(i)

[1]

Calculate an expression for $C'(t)$ .

Part (b)(ii)

[1]

Evaluate $C'(1)$ .

Part (b)(iii)

[1]

Interpret the meaning of $C'(1)$ in context.

Part (c)

[1]

Represent the tangent line to the graph of $C$ at $t=1$ by an equation.

Part (d)(i)

[2]

Determine all values of $t$ in $0<t<4$ at which the graph has a horizontal tangent. Show the equation solved.

Part (d)(ii)

[1]

Explain how the graph supports the two numerical answers from part (d)(i).

Question 4

GRAPHING CALCULATOR NOT PERMITTED

Let $f(x)=x^2-3x+1$ . What is the average rate of change of $f$ on the interval $[1,4]$ ?

A. $-2$

B. $2$

C. $3$

D. $6$

[1]

Select Answer

Question 5

Graphing calculator not permitted

The function $f$ is defined on $[-3,4]$ by

$f(x)= \begin{cases} x+2, & -3\leq x<0,\ 1+\dfrac{x^2}{2}, & 0\leq x\leq2,\ x+1, & 2<x\leq4. \end{cases}$

Its graph is shown.

Part (a)

[3]

Determine whether $f$ is continuous at $x=0$ . Justify your conclusion using limits and the function value.

Part (b)

[3]

Determine whether $f$ is differentiable at $x=2$ . Justify your conclusion using continuity and one-sided derivatives.

Part (c)(i)

[1]

Calculate $f'(1)$ .

Part (c)(ii)

[1]

Represent the line tangent to the graph of $f$ at $x=1$ by an equation.

Part (d)

[1]

Explain why $f$ must be continuous at $x=1$ .

Question 6

Graphing calculator not permitted

Let $f(x)=x^2$ . The point $P=(1,1)$ lies on the graph. For each nonzero value of $h$ , the point

$Q_h=(1+h,f(1+h))$

also lies on the graph. Several secant lines through $P$ and $Q_h$ , together with the tangent line at $P$ , are shown.

Part (a)

[3]

Calculate the slope of the secant line through $P$ and $Q_h$ for each of the following values: $h=1$ , $h=0.5$ , and $h=-0.5$ .

Part (b)

[2]

Evaluate $f'(1)$ by using the limit definition of the derivative.

Part (c)

[2]

Represent the tangent line to the graph of $f$ at $P$ by an equation.

Part (d)

[2]

Explain how the secant lines shown support the result from part (b). Your explanation must distinguish a secant slope from the tangent slope.

Question 7

Graphing calculator not permitted

The graph shows differentiable functions $u$ and $v$ . The solid curve represents $u$ , and the dashed line represents $v$ . The dotted line is tangent to the graph of $u$ at $x=1$ . All necessary function values and slopes can be determined from the coordinate grid.

Part (a)

[4]

Determine each of the following values from the graph:

$u(1)$ , $v(1)$ , $u'(1)$ , and $v'(1)$ .

Let $p(x)=u(x)v(x)$ .

Part (b)

[2]

Calculate $p'(1)$ .

Let $q(x)=\dfrac{u(x)}{v(x)}$ .

Part (c)

[2]

Calculate $q'(1)$ .

Part (d)

[1]

Represent the line tangent to the graph of $p$ at $x=1$ by an equation.

Question 8

Graphing calculator not permitted

The continuous function $f$ is piecewise linear on $[-3,4]$ . Its graph consists of line segments joining the labelled points shown.

Part (a)

[2]

Calculate the average rate of change of $f$ on each interval:

$[-3,-1]$
$[1,4]$

Part (b)

[4]

Represent $f'(x)$ on $-3<x<4$ by giving a piecewise definition or an accurately labelled graph. Indicate all points where $f'$ is undefined.

Part (c)

[2]

Determine the interior values of $x$ at which $f'(x)$ does not exist. Explain your reasoning.

Part (d)

[1]

Interpret the statement $f'(x)=0$ for $1<x<3$ in terms of the behaviour of $f$ .

Question 9

Graphing calculator not permitted

Let $f(x)=\sqrt[3]{x}$ . The graph of $f$ and two secant lines through the origin are shown. One secant uses $h=1$ , and the other uses $h=\frac18$ .

Part (a)

[2]

Verify that $f$ is continuous at $x=0$ .

Part (b)

[3]

Determine whether $f'(0)$ exists as a finite real number. Use the limit definition of the derivative to justify your conclusion.

Part (c)

[2]

Explain why this function demonstrates that continuity at a point does not guarantee differentiability at that point.

Part (d)

[2]

Represent each of the two secant lines shown by an equation.

Question 10

Graphing calculator not permitted

The graph of $f$ is shown. Which statement is true?

A. $f$ is differentiable at $x=-1$ because both visible pieces have finite slopes.

B. $f$ is continuous but not differentiable at $x=1$ .

C. $f$ is differentiable at $x=1$ because it is continuous there.

D. $f$ is continuous at $x=-1$ because $\displaystyle\lim_{x\to-1}f(x)$ exists.

[1]

Select Answer

Practice Questions

2. Differentiation: Definition and Fundamental Properties

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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