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SL 5.3—Introduction to derivatives - IB Questionbank | RevisionDojo

Practice SL 5.3—Introduction to derivatives with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A local bakery is analyzing the profit from its signature pastry, which is modeled by the function $f(x) = x^3 - 6x^2 + 9x + 2$ , where $x$ represents the number of hours after opening and $f(x)$ represents the profit in hundreds of dollars.

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit with respect to time.

[3]

Find the critical points of the function by solving $f^{\prime}(x) = 0$ . These points will help the bakery understand when the profit is at a local maximum or minimum.

[4]

Determine whether the profit is increasing or decreasing on the intervals defined by the critical points.

[4]

Identify any local maximum or minimum points and justify your answer.

[4]

Question 2

SLPaper 2

A technology company is analyzing the performance of its new software product. The function representing the profit from the software is given by:

$f(x)=2 x^3-9 x^2+12 x+1$

Find the derivative $f^{\prime}(x)$ .

[2]

Solve for the critical points where $f^{\prime}(x)=0$ .

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 3

SLPaper 2

A student is studying the decay of a chemical reactant over time in a laboratory experiment. The concentration of the reactant can be modeled by the exponential decay equation: $C(t) = C_0 e^{-k t}$ where $C(t)$ is the concentration at time $t$ , $C_0$ is the initial concentration, and $k$ is a positive constant representing the decay rate.

Show that the rate of change of concentration, $\frac{d C}{d t}$ , is always negative.

[3]

Evaluate the limit of $C(t)$ as $t \rightarrow \infty$ and interpret the result.

[3]

Sketch the graph of $C(t)$ against $t$ . Label the key features, including the initial concentration $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[4]

Find the half-life of the concentration, $t_{\frac{1}{2}}$ , in terms of $k$ .

[3]

Rewrite the equation in the form $\ln C(t) = \ln C_0 - k t$ and identify the gradient and intercept of the line.

[3]

Use the rewritten equation to determine the values of $C_0$ and $k$ if the data plotted shows a straight line with slope $-0.7$ and intercept $2$ .

[4]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

SLPaper 1

A bakery's daily profit, $h(x)$ , in hundreds of dollars, is modeled by the function $h(x) = -4x^5 + 15x^3 + 2$ , where $x \ge 0$ is the production level in hundreds of units.

Calculate the derivative $h'(x)$ to find the rate of change of profit with respect to the production level $x$ .

[2]

Determine the critical points of $h(x)$ for $x \ge 0$ by setting $h'(x) = 0$ .

[4]

Use the second derivative test to classify the nature of the critical points found in Part 2. This helps the bakery decide whether to adjust production levels.

[4]

State the intervals where the profit function is increasing and where it is decreasing for $x \ge 0$ .

[2]

Question 6

SLPaper 2

A startup company is analyzing its profit growth over time to make strategic decisions. The rate at which the company's profit is changing over time is given by:

$\frac{d P}{d t}=3 t^2-8 t$

where $P$ represents the company's profit (in thousands of dollars), and $t$ is the time in years since the company was founded ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context of the company's profit.

[3]

Question 7

HLPaper 2

Let $f(x) = \frac{2x - 3}{x + 1}$

Evaluate the limit:

$\lim_{x \to \infty} \frac{2x - 3}{x + 1}$

[3]

Find the derivative ( $f'(x)$ )

[5]

Determine the value of $f'(1)$ .

[1]

State the interpretation of $f'(1)$ in terms of the rate of change of $f(x)$ .

[2]

Question 8

SLPaper 1

The surface area of an open box with a volume of $32 \text{ cm}^3$ and a square base with sides of length $x$ cm is given by $S(x) = x^2 + \frac{128}{x}$ where $x > 0$ .

Find $S'(x)$ .

[3]

Solve $S'(x) = 0$ .

[2]

Interpret your answer to part 2 in context.

[1]

Question 9

SLPaper 1

A tech company is analyzing the performance of a new software application, represented by the function $g(x) = 5x^3 - 12x^2 + 6x + 10$ , where $x$ is the number of updates released.

Calculate the derivative $g^{\prime}(x)$ to find the rate of change of performance with respect to updates.

[2]

Find the critical points of $g(x)$ to identify when the performance is at a stationary point.

[4]

Determine whether these points correspond to local maxima or minima using the second derivative test.

[3]

Find the instantaneous rate of change of the function at $x = 2$ .

[3]

Using your result from Part 2, explain what these values of $x$ tell us about the slope of the function.

[4]

Question 10

SLPaper 1

Consider the function $g(x) = \frac{x^4}{4}$.

Find $g'(x)$.

[1]

Find the $x$-coordinate of the point at which the normal to the graph of $g$ has gradient $-\frac{1}{8}$.

[3]

Find the gradient of the graph of $g$ at $x = -\frac{1}{2}$.

[2]

Paper

Level

Question 1

SLPaper 1

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit with respect to time.

[3]

Find the critical points of the function by solving $f^{\prime}(x) = 0$ . These points will help the bakery understand when the profit is at a local maximum or minimum.

[4]

Determine whether the profit is increasing or decreasing on the intervals defined by the critical points.

[4]

Identify any local maximum or minimum points and justify your answer.

[4]

Question 2

SLPaper 2

A technology company is analyzing the performance of its new software product. The function representing the profit from the software is given by:

$f(x)=2 x^3-9 x^2+12 x+1$

Find the derivative $f^{\prime}(x)$ .

[2]

Solve for the critical points where $f^{\prime}(x)=0$ .

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 3

SLPaper 2

Show that the rate of change of concentration, $\frac{d C}{d t}$ , is always negative.

[3]

Evaluate the limit of $C(t)$ as $t \rightarrow \infty$ and interpret the result.

[3]

Sketch the graph of $C(t)$ against $t$ . Label the key features, including the initial concentration $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[4]

Find the half-life of the concentration, $t_{\frac{1}{2}}$ , in terms of $k$ .

[3]

Rewrite the equation in the form $\ln C(t) = \ln C_0 - k t$ and identify the gradient and intercept of the line.

[3]

Use the rewritten equation to determine the values of $C_0$ and $k$ if the data plotted shows a straight line with slope $-0.7$ and intercept $2$ .

[4]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

SLPaper 1

A bakery's daily profit, $h(x)$ , in hundreds of dollars, is modeled by the function $h(x) = -4x^5 + 15x^3 + 2$ , where $x \ge 0$ is the production level in hundreds of units.

Calculate the derivative $h'(x)$ to find the rate of change of profit with respect to the production level $x$ .

[2]

Determine the critical points of $h(x)$ for $x \ge 0$ by setting $h'(x) = 0$ .

[4]

Use the second derivative test to classify the nature of the critical points found in Part 2. This helps the bakery decide whether to adjust production levels.

[4]

State the intervals where the profit function is increasing and where it is decreasing for $x \ge 0$ .

[2]

Question 6

SLPaper 2

A startup company is analyzing its profit growth over time to make strategic decisions. The rate at which the company's profit is changing over time is given by:

$\frac{d P}{d t}=3 t^2-8 t$

where $P$ represents the company's profit (in thousands of dollars), and $t$ is the time in years since the company was founded ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context of the company's profit.

[3]

Question 7

HLPaper 2

Let $f(x) = \frac{2x - 3}{x + 1}$

Evaluate the limit:

$\lim_{x \to \infty} \frac{2x - 3}{x + 1}$

[3]

Find the derivative ( $f'(x)$ )

[5]

Determine the value of $f'(1)$ .

[1]

State the interpretation of $f'(1)$ in terms of the rate of change of $f(x)$ .

[2]

Question 8

SLPaper 1

The surface area of an open box with a volume of $32 \text{ cm}^3$ and a square base with sides of length $x$ cm is given by $S(x) = x^2 + \frac{128}{x}$ where $x > 0$ .

Find $S'(x)$ .

[3]

Solve $S'(x) = 0$ .

[2]

Interpret your answer to part 2 in context.

[1]

Question 9

SLPaper 1

A tech company is analyzing the performance of a new software application, represented by the function $g(x) = 5x^3 - 12x^2 + 6x + 10$ , where $x$ is the number of updates released.

Calculate the derivative $g^{\prime}(x)$ to find the rate of change of performance with respect to updates.

[2]

Find the critical points of $g(x)$ to identify when the performance is at a stationary point.

[4]

Determine whether these points correspond to local maxima or minima using the second derivative test.

[3]

Find the instantaneous rate of change of the function at $x = 2$ .

[3]

Using your result from Part 2, explain what these values of $x$ tell us about the slope of the function.

[4]

Question 10

SLPaper 1

Consider the function $g(x) = \frac{x^4}{4}$.

Find $g'(x)$.

[1]

Find the $x$-coordinate of the point at which the normal to the graph of $g$ has gradient $-\frac{1}{8}$.

[3]

Find the gradient of the graph of $g$ at $x = -\frac{1}{2}$.

[2]

Paper

Level

Question 1

SLPaper 1

Calculate the derivative $f^{\prime}(x)$ to determine the rate of change of the profit with respect to time.

[3]

Find the critical points of the function by solving $f^{\prime}(x) = 0$ . These points will help the bakery understand when the profit is at a local maximum or minimum.

[4]

Determine whether the profit is increasing or decreasing on the intervals defined by the critical points.

[4]

Identify any local maximum or minimum points and justify your answer.

[4]

Question 2

SLPaper 2

A technology company is analyzing the performance of its new software product. The function representing the profit from the software is given by:

$f(x)=2 x^3-9 x^2+12 x+1$

Find the derivative $f^{\prime}(x)$ .

[2]

Solve for the critical points where $f^{\prime}(x)=0$ .

[3]

Determine the intervals where the function is increasing and decreasing.

[3]

Determine the $x$ -coordinates of the local maximum and minimum points.

[2]

Sketch the graph of the function, clearly labeling the critical points and behavior.

[3]

Question 3

SLPaper 2

Show that the rate of change of concentration, $\frac{d C}{d t}$ , is always negative.

[3]

Evaluate the limit of $C(t)$ as $t \rightarrow \infty$ and interpret the result.

[3]

Sketch the graph of $C(t)$ against $t$ . Label the key features, including the initial concentration $C_0$ .

In the following graph, $C_0=4$ and $k=0.4$ as an example:

[4]

Find the half-life of the concentration, $t_{\frac{1}{2}}$ , in terms of $k$ .

[3]

Rewrite the equation in the form $\ln C(t) = \ln C_0 - k t$ and identify the gradient and intercept of the line.

[3]

Use the rewritten equation to determine the values of $C_0$ and $k$ if the data plotted shows a straight line with slope $-0.7$ and intercept $2$ .

[4]

Question 4

SLPaper 2

Note: In this question, distance is in metres and time is in seconds.

A particle P moves in a straight line for five seconds. Its acceleration at time $t$ is given by $a = 3t^2 - 14t + 8$, for $0 \le t \le 5$.

When $t = 0$, the velocity of P is $3 \text{ m s}^{-1}$.

Write down the values of $t$ when $a = 0$.

[2]

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

[2]

Find an expression for the velocity of P at time $t$.

[6]

Find the total distance travelled by P when its velocity is increasing.

[4]

Question 5

SLPaper 1

A bakery's daily profit, $h(x)$ , in hundreds of dollars, is modeled by the function $h(x) = -4x^5 + 15x^3 + 2$ , where $x \ge 0$ is the production level in hundreds of units.

Calculate the derivative $h'(x)$ to find the rate of change of profit with respect to the production level $x$ .

[2]

Determine the critical points of $h(x)$ for $x \ge 0$ by setting $h'(x) = 0$ .

[4]

Use the second derivative test to classify the nature of the critical points found in Part 2. This helps the bakery decide whether to adjust production levels.

[4]

State the intervals where the profit function is increasing and where it is decreasing for $x \ge 0$ .

[2]

Question 6

SLPaper 2

A startup company is analyzing its profit growth over time to make strategic decisions. The rate at which the company's profit is changing over time is given by:

$\frac{d P}{d t}=3 t^2-8 t$

where $P$ represents the company's profit (in thousands of dollars), and $t$ is the time in years since the company was founded ( $t \ge 0$ ).

Find the values of $t$ for which the rate of change of profit is zero.

[4]

Determine the intervals of time for which the profit is increasing and decreasing.

[4]

Interpret the meaning of these intervals in the context of the company's profit.

[3]

Question 7

HLPaper 2

Let $f(x) = \frac{2x - 3}{x + 1}$

Evaluate the limit:

$\lim_{x \to \infty} \frac{2x - 3}{x + 1}$

[3]

Find the derivative ( $f'(x)$ )

[5]

Determine the value of $f'(1)$ .

[1]

State the interpretation of $f'(1)$ in terms of the rate of change of $f(x)$ .

[2]

Question 8

SLPaper 1

The surface area of an open box with a volume of $32 \text{ cm}^3$ and a square base with sides of length $x$ cm is given by $S(x) = x^2 + \frac{128}{x}$ where $x > 0$ .

Find $S'(x)$ .

[3]

Solve $S'(x) = 0$ .

[2]

Interpret your answer to part 2 in context.

[1]

Question 9

SLPaper 1

A tech company is analyzing the performance of a new software application, represented by the function $g(x) = 5x^3 - 12x^2 + 6x + 10$ , where $x$ is the number of updates released.

Calculate the derivative $g^{\prime}(x)$ to find the rate of change of performance with respect to updates.

[2]

Find the critical points of $g(x)$ to identify when the performance is at a stationary point.

[4]

Determine whether these points correspond to local maxima or minima using the second derivative test.

[3]

Find the instantaneous rate of change of the function at $x = 2$ .

[3]

Using your result from Part 2, explain what these values of $x$ tell us about the slope of the function.

[4]

Question 10

SLPaper 1

Consider the function $g(x) = \frac{x^4}{4}$.

Find $g'(x)$.

[1]

Find the $x$-coordinate of the point at which the normal to the graph of $g$ has gradient $-\frac{1}{8}$.

[3]

Find the gradient of the graph of $g$ at $x = -\frac{1}{2}$.

[2]

SL 5.3—Introduction to derivatives Questionbank