Calculus — Free Maths AI HL Practice Questions

Find the general solution of the differential equation. [2 marks]

Given that the population is $5000$ at $t = 0$, find the population after $10$ years, correct to the nearest hundred. [2 marks]

Hence, by solving the differential equation, show that $$P = \frac{1000}{1 + 4e^{-5t}}.$$ [3 marks]

Hence, determine the particular solution satisfying $V(0) = 10$. [2 marks]

Show that the particular solution found in (b) approaches the line $V = t + 1$ asymptotically as $t \to \infty$. Hence analyse what this behaviour implies about the effectiveness of the filtering process over time. [2 marks]

Find the particular solution of the differential equation, expressing $V^2$ in terms of $t$. [2 marks]

Find the time, in minutes, when the volume of gas reaches $10\,\text{cm}^3$. Give your answer correct to 3 significant figures. [1 mark]

The model predicts that the volume of gas increases without bound as $t \to \infty$. Assess the validity of this prediction in the context of the experiment, referring to the rate of change $\dfrac{dV}{dt}$ as $V$ increases. [1 mark]

Find $\displaystyle\lim_{t \to \infty} N(t)$. [2 marks]

Explain what the value found in part (b) represents in the context of this model, and justify whether the population ever actually reaches this value. [2 marks]

Given that $g(x) = \dfrac{[f(x)]^2 - 1}{x - 2}$, determine $\displaystyle\lim_{x \to 2} g(x)$. [2 marks]

The scientist defines $F(1) = 1$ to extend $F$ to all $x$. Using the formal definition of continuity, verify that the extended function is continuous at $x = 1$. [1 mark]

Using the limit definition of the derivative, $$F'(1) = \lim_{h \to 0} \frac{F(1+h) - F(1)}{h}$$ determine whether the extended function is differentiable at $x = 1$. Justify your answer fully. [4 marks]

Find $\lim_{x \to 3} f(x)$ by first factorising the numerator. [3 marks]

A new function $g$ is defined as $$g(x) = \begin{cases} \dfrac{x^2 - 9}{x - 3} & x \neq 3 \\[6pt] k & x = 3 \end{cases}$$ Determine the value of $k$ for which $g$ is continuous at $x = 3$, and justify whether $g$ and $f$ represent the same function. [3 marks]

Find the rate of change of volume at $t = 5$ minutes, and state whether the volume is increasing or decreasing at this instant. [2 marks]

Determine the time at which the volume in the tank is greatest, and find this maximum volume. Justify your answer. [3 marks]

Find the time at which the velocity is $37\,\text{m s}^{-1}$. [2 marks]

Determine whether the particle ever moves in the negative direction. Justify your answer. [1 mark]

(i) Show that $f$ is an odd function. [1]
(ii) Hence determine the gradient of the tangent to the graph of $f$ at $x = 1$, justifying your reasoning. [2]
(iii) Write down the equation of the tangent to the graph of $f$ at $x = 1$. [1 mark]

Find the equation of the tangent to the curve at $P$, giving your answer in the form $y = mx + c$. [2 marks]

The tangent at $P$ intersects the $x$-axis at point $R$. Determine the $x$-coordinate of $R$. [1 mark]

Find $V(t)$, given that the tank initially contains $50$ litres of water. [4 marks]

Determine the maximum volume of water in the tank and justify that is a maximum. [3 marks]

Calculate the total distance travelled by the particle in the first $4$ seconds. [3 marks]

Calculate the mass of the colony after $4$ hours, correct to 3 significant figures. [2 marks]

A second biologist claims this model predicts that the colony's mass will eventually exceed any fixed value, no matter how large. Evaluate this claim by analysing the behaviour of $m(t)$ as $t \to \infty$, and comment on whether the model is realistic for large $t$. [2 marks]

Find an expression for $v(t)$. [3 marks]

Determine the value of $t$ at which the particle first reaches its minimum velocity, and state that minimum velocity. [1 mark]

Given that the initial population is 300 birds, solve the differential equation to find an expression for $P$ in terms of $t$. Give your answer in the form $P = \dfrac{A}{1 + Be^{-kt}}$, where $A$, $B$, $k$ are constants. [3 marks]

Calculate the population after 10 years, correct to the nearest whole number. [1 mark]

Given that $x = 0$ when $t = 0$, and $x = 5$ when $t = 2$, calculate the value of $k$, giving your answer correct to 3 significant figures. [4 marks]

Calculate the time at which the concentration reaches $20\,\text{g}\,\text{l}^{-1}$, giving your answer in hours and minutes, correct to the nearest minute. [4 marks]

Determine the value of $k$, giving your answer correct to $3$ significant figures. [2 marks]

Calculate the time, in hours, for the number of bacteria to reach $5000$. Give your answer correct to the nearest hour. [2 marks]

Determine $\displaystyle\lim_{t \to \infty} T(t)$ and explain what this value represents in the context of the model. [2 marks]

The heating element is designed to switch off when the temperature reaches $95\,^\circ\text{C}$. Solve $T(t) = 95$ algebraically to find the time at which this occurs, giving your answer in minutes correct to 3 significant figures. [2 marks]

Hence calculate $\lim_{x \to 2} f(x)$. [1 mark]

A student proposes extending $f$ to a new function $g$ defined on all of $\mathbb{R}$ by $$g(x) = \begin{cases} x^2 + 2x + 4, & x \neq 2 \\ k, & x = 2 \end{cases}$$ Determine the value of $k$ for which $g$ is continuous at $x = 2$, and justify your answer. [3 marks]

A second function is defined by $$g(x) = \begin{cases} \dfrac{x^2 - 4}{(x-2)^2}, & x \neq 2 \\[6pt] 3, & x = 2 \end{cases}$$ Determine whether $\displaystyle\lim_{x \to 2} g(x)$ exists. [1 mark]

Using your results from parts (a) and (b), compare the type of discontinuity at $x = 2$ for $f$ (when $k \neq 4$) and for $g$. Justify your answer by describing the behaviour of each function near $x = 2$. [2 marks]

State three conditions required for $g$ to be continuous at $x = 1$, and determine, with justification, whether each condition is satisfied. [2 marks]

Hence classify the discontinuity of $g$ at $x = 1$. Explain whether redefining $g(1)$ could make $g$ continuous at that point, and state the value that would be required. [2 marks]

The tangent line at $P$ has equation $y = x + 2$. Determine the $x$-coordinate of the point where this tangent line intersects the line $y = 3x - 4$. [2 marks]

Calculate the average rate of change of $f$ between $x = -2$ and $x = 4$. Give your answer as an exact fraction. [2 marks]

Find $A'(x)$ and hence calculate the rate of change of area with respect to width when $x = 10$. State the meaning of this value in context. [2 marks]

Determine the value of $x$ that maximises the enclosed area. Justify that this value gives a maximum and not a minimum. [2 marks]

Calculate the value of $t$ at which the rate of change of volume is $-40$ litres per second. Give your answer correct to 3 significant figures. [3 marks]

Calculate the acceleration of the particle at $t = 2$ seconds. [2 marks]

The particle changes direction once for $t > 0$. Determine the time at which this occurs and find the position of the particle at that instant. [2 marks]

Given that the profit in 2020 was USD 5 million, determine the particular solution for $P(t)$. [2 marks]

The company claims this model predicts long-term profit growth. By finding the value of $t$ at which $P(t)$ is maximised and evaluating $P$ at that point, determine whether this claim is valid. [3 marks]

Determine the displacement of the particle from the origin at $t = 2$ seconds. Give your answer correct to 3 significant figures. [2 marks]

Find the total distance travelled by the particle in the interval $0 \leq t \leq 2$. Give your answer correct to 3 significant figures. (Note: total marks = 8; adjust paper allocation accordingly.) [3 marks]

Calculate the total distance travelled by the particle in the first 3 seconds. [3 marks]

The sheet is cut vertically at $x = k$, where $0 < k < 10$, so that the left portion has exactly half the total cross-sectional area. Determine the value of $k$. [3 marks]

After 3 hours, the concentration is measured as $12\,\text{mg\,L}^{-1}$. (i) Calculate the value of $k$, giving your answer correct to three significant figures. [2 marks]
(ii) A concentration of at least $19.9\,\text{mg\,L}^{-1}$ is required for the drug to be therapeutically effective. Determine the time, to the nearest minute, at which this threshold is first reached. [2 marks]
(iii) The drug is considered unsafe if the concentration ever reaches $20\,\text{mg\,L}^{-1}$. Using your model, evaluate whether the drug can be administered safely for an indefinite period. Justify your answer with reference to the behaviour of $C(t)$ as $t \to \infty$. [1 mark]

After 5 minutes, 4 grams of substance A remain. (i) Calculate the exact value of $k$. [2 marks]
(ii) A chemist claims that, under this model, the reaction will consume at least $9.5$ grams of substance A within the first 30 minutes. Determine whether this claim is correct, justifying your answer with appropriate calculations. [3 marks]

Show that $N'(t) = \dfrac{-24t^2 + 10t + 5}{(2t^2+1)^2}$ and hence evaluate $N'(0)$. [3 marks]

Interpret the value of $N'(0)$ in the context of the model. [1 mark]

By considering $N'(t)$ and the limit found in part (a), justify whether the model predicts that the bacterial population ever reaches or exceeds $6.5$ thousand. [3 marks]

Evaluate $\displaystyle\lim_{x \to 0} g(x)$, justifying your answer by explicit use of the standard result $\displaystyle\lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1$. [3 marks]

(i) Given that $g$ is not defined at $x = 0$, state the value that must be assigned to $g(0)$ for $g$ to be continuous at $x = 0$. [1 mark]
(ii) Let $h(x) = f(x) \cdot g(x)$ for $x \neq 0$. Determine $\displaystyle\lim_{x \to 0} h(x)$ and hence determine the value of $h(0)$ that makes $h$ continuous at $x = 0$. Justify whether this value equals $f(0) \cdot g(0)$, where $g(0)$ is the value found in (c)(i), and state what general property of continuous functions this illustrates. [3 marks]

Given that $h = 2$ and $\frac{dh}{dt} = 0.05$ when $t = 0$, find the value of $k$. [2 marks]

Solve the differential equation from part (a) subject to the initial condition $h = 2$ when $t = 0$, expressing $h$ as a function of $t$. [3 marks]

Hence determine the limiting depth of water as $t \to \infty$, and justify whether this depth is ever reached for any finite value of $t$. [2 marks]

Hence, find the values of $t$ at which the particle is instaneously at rest. [2 marks]

By considering the sign of $v(t)$ on either side of each value found in part (b), determine the nature of the corresponding turning points on the graph of $s(t)$, and state the direction of motion of the particle on the interval $1 < t < 3$. [3 marks]

Calculate the total distance travelled by the particle in the first $4$ seconds, and hence determine whether the particle's displacement from $O$ at $t = 4$ is greater or less than the total distance it has travelled. Justify your answer. (HL extension) [4 marks]

Calculate the average rate of production over the 12-hour period. Give your answer in kilograms per hour, correct to 3 significant figures. [2 marks]

The reactor must be cleaned if the total production in any 12-hour batch falls below 20 kg. Without using a calculator to evaluate $e^{-2.4}$ numerically, show algebraically that cleaning is not required for this batch. Hence state the minimum integer value of the threshold, in kilograms, for which the cleaning decision could not be determined without a calculator. [3 marks]

Determine the time, in years, at which the population first falls to 25 hundred fish. [2 marks]

The biologist claims the population will eventually stabilise above zero and never become extinct under this model. By finding $\displaystyle\lim_{t \to \infty} P(t)$ and analysing whether $P(t) = 0$ has a solution for $t \geq 0$, evaluate this claim. [3 marks]