Calculus

1. Limits (HL): practice question 1 for Calculus?

1. Limits (HL): practice question 2 for Calculus?

1. Limits (HL): practice question 3 for Calculus?

1. Limits (HL): practice question 4 for Calculus?

1. Limits (HL): practice question 5 for Calculus?

1. Limits (HL): practice question 6 for Calculus?

1. Limits (HL): practice question 7 for Calculus?

1. Limits (HL): practice question 8 for Calculus?

1. Limits (HL): practice question 9 for Calculus?

1. Limits (HL): practice question 10 for Calculus?

The radius of a circular oil spill is $r(t) = 2t + 5$ cm. Find the rate of change of the radius with respect to time.

Find the derivative of $f(x) = \frac{4x^3 - 2x^2}{x}$.

At which point does the curve $y = x^2 - x$ have a gradient of $5$?

The total cost of producing $x$ kilograms of tea is $C(x) = 4\sqrt{x} + 20$. Find the marginal cost when $x = 16$.

A particle moves such that its position is $s(t) = t^3 - 6t^2 + 9t$. At what time(s) is the particle at rest?

The total surface area of a cube is $S = 6x^2$, where $x$ is the side length. Find the rate of change of surface area with respect to $x$ when $x = 4$ cm.

The value of a car $V$ in dollars after $t$ years is $V(t) = 25000(0.8)^t$. Find the rate of change of the car's value when $t = 2$.

The value of a car depreciates according to $V(t) = 25000 - 4000t + 200t^2$, where $t$ is in years. Find the rate of change of value after $5$ years.

Find the equation of the normal to $y = x^3 - 3x + 2$ at the point $(2, 4)$.

The gradient of the curve $y = \frac{k}{x} + x$ at $x = 2$ is $0.75$. Find the value of $k$.

The position of an object is $s(t) = 20 - 5t$ meters. Find its velocity.

If $f(x) = 2x - 3x^2$, find $f'(2)$.

Find the derivative of $y = \frac{x^2 + x}{x}$ for $x \neq 0$.

The gradient of a tangent to $y = x^2$ is $10$. Find the $x$-coordinate of the point of tangency.

If $f(x) = (x+2)(x-3)$, find $f'(x)$.

The gradient of the tangent to $y = x^2 + kx$ at $x = 3$ is 10. Find $k$.

The height of a weather balloon is $H(t) = -0.1t^3 + 3t^2 + 10t$. Find the rate of change of the height at $t = 10$.

Determine the $y$-intercept of the tangent line to $f(x) = \ln(x)$ at the point where $x = e$.

A stone is dropped into a pond, creating a circular ripple. The radius $r$ increases at $20$ cm/s. Find the rate at which the area $A$ of the ripple is increasing when the radius is $50$ cm.

The gradient of the curve $y = f(x)$ at any point $(x, y)$ is given by $3x^{2} - 4x$. If the curve passes through $(2, 5)$, find the equation of the tangent at that point.

Find the gradient of the tangent to the curve $f(x) = \frac{1}{2}x^2 + 4x$ at $x = -2$.

At what $x$-coordinate is the gradient of the tangent to $y = x^2$ equal to the gradient of the tangent to $y = 6x - 5$?

Find the gradient of the normal to the curve $y = x^3 - x$ at $x = 0$.

The point $P(1, 2)$ lies on a curve. If the gradient of the tangent at $P$ is 3, find the equation of the tangent.

Find the equation of the tangent to $y = x^3 - 2x^2 + 5x - 7$ at the point where $x = 1$.

A path is modeled by $y = 4 - x^2$. Find the gradient of the normal where the path crosses the $x$-axis for $x > 0$.

Find the $x$-intercept of the tangent to the curve $y = x^2 - x$ at the point $(2, 2)$.

Find the product of the gradients of the tangents to the curve $y = x^2 - 4x + 3$ at its $x$-intercepts.

A curve has equation $y = x^3 - px$. The tangent at $x = 1$ is perpendicular to the line $y = 0.5x + 3$. Find the value of $p$.

The tangent to the curve $y = x^2 + c$ at $x = 3$ intersects the $x$-axis at $x = 1$. Find the value of $c$.

A piece of wire 100 cm long is cut into two pieces. One piece is bent into a square. If the side of the square is $x$, the remaining wire is $100 - 4x$. If this is bent into another square, the total area is $A = x^2 + (25 - x)^2$. Find the value of $x$ that minimizes the total area.

The volume of a cone with slant height 10 is $V = \frac{1}{3}\pi(100 - h^2)h$. Use a GDC to find the height $h$ that maximizes the volume (to 2 decimal places).

A company finds its profit function is $P(x) = -2x^2 + 120x - 1000$. What is the maximum profit?

A rectangular enclosure is built with 80 meters of wire. The area is $A = x(40 - x)$. Find the value of $x$ that maximizes the area.

A gift box with a square base $x$ and height $h$ has a total surface area of 600 cm$^2$. The volume is given by $V(x) = 150x - 0.25x^3$. Find the value of $x$ that maximizes the volume.

A particle moves along a line such that its position is $s(t) = 2t^3 - 15t^2 + 24t$. Find the time $t$ at which the particle's velocity is a minimum.

The strength of a beam is proportional to $S = w h^2$, where $w$ is the width and $h$ is the height. If $w^2 + h^2 = 300$, then $S(w) = w(300 - w^2) = 300w - w^3$. Find the width $w$ that maximizes the strength.

A gardener wants to create a flower bed in the shape of a sector of a circle with a fixed area of 50 m$^2$. Find the radius $r$ in meters that minimizes the perimeter of the bed.

A rectangle is inscribed in an ellipse with the equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$. Find the maximum area of such a rectangle.

An open-top water tank with a square base is to have a volume of 32 m$^3$. Find the minimum surface area in m$^2$ of the tank.

The velocity of a car is $v(t) = 2t + 1$. Find the time when the velocity is $15$ m/s.

A particle moves with velocity $v(t) = \frac{1}{t}$. Find the displacement between $t = 1$ and $t = e$.

The acceleration of a particle is $a(t) = 6$. If $v(1) = 10$, find the initial velocity $v(0)$.

If velocity is $v(t) = \cos(t)$, find the displacement from $t = 0$ to $t = \frac{\pi}{2}$.

A car's position is $s(t) = 50t - 2t^2$. Find the distance traveled until it comes to a stop.

An object's velocity is $v(t) = 4t^3$. Find the average acceleration from $t = 1$ to $t = 2$.

The displacement of a particle is $s(t) = \sin(t) + \cos(t)$. Find the velocity at $t = \frac{\pi}{4}$.

A particle moves such that its velocity $v$ at time $t$ is $v(t) = 10 \sin(t) - 2$. Find the total distance traveled from $t=0$ to $t=\pi$.

A drone's height is $h(t) = \frac{1}{3}t^3 - 4t^2 + 12t$. At what time $t$ is the vertical velocity a minimum?

The velocity of a particle is $v(t) = e^{0.5t} - 3$. At what time $t$ does the particle change direction?

Estimate ( \int_{0}^{4} f(x) , dx ) using the trapezoidal rule with ( h = 2 ) and the following table: ( x=0, f(x)=1 ); ( x=2, f(x)=3 ); ( x=4, f(x)=5 ).

Which of the following is an antiderivative of ( f(x) = 3x^2 + 2x )?

The area under a constant function ( f(x) = 5 ) from ( x = 2 ) to ( x = 6 ) is:

If ( f'(x) = \frac{1}{x} ) and ( f(1) = 2 ), find ( f(e) ).

Calculate the volume generated by rotating the region bounded by $y = x^3$, $y = 0$, and $x = 2$ about the $x$-axis.

The velocity of a particle is $v(t) = 6t - t^2$ m/s. Find the total distance traveled from $t = 0$ to $t = 8$.

Find the average value of $f(x) = x^2$ on the interval $[0, 3]$.

A particle's velocity is $v(t) = 12t - 3t^2$. Find the distance traveled between the two times the particle is at rest.

Find the area of the region bounded by $y = x^2 - 6x + 8$ and $y = -x^2 + 6x - 8$.

The gradient of a curve is given by $\frac{dy}{dx} = 2x - 5$. If the area between the curve, the $x$-axis, and the lines $x=1$ and $x=3$ is to be calculated, and the curve lies entirely below the $x$-axis in this interval, which of the following is the area given $y(2)=-5$?

A speed-time graph is a straight line from $(0,0)$ to $(10,20)$. Use integration to find the total distance traveled from $t = 0$ to $t = 10$.

Find the area bounded by $y = \frac{1}{x+2}$, the $x$-axis, $x = 0$, and $x = 2$.

The marginal profit of a firm is $P'(x) = 20 - 0.1x$. Calculate the change in profit when production increases from $x = 50$ to $x = 100$.

Determine the area under $y = e^x + 1$ from $x = 0$ to $x = 1$.

Find the area of the region bounded by $y = 2x$, the x-axis, $x = 1$, and $x = 3$.

Estimate $\int_1^5 \ln(x) dx$ using the trapezoidal rule with 2 intervals of equal width.

A particle moves with velocity $v(t) = e^{0.1t}$. Find the distance traveled from $t = 0$ to $t = 10$.

A curve is given by $y = x^2 - 6x + 8$. Find the area of the region bounded by the curve and the x-axis.

Find the area of the region bounded by $y = \frac{1}{x^2}$, the x-axis, $x=1$, and $x=2$.

Find the area of the region bounded by $y = e^{0.2x}$, the x-axis, the y-axis, and $x=5$.

1. Differential equations (HL): practice question 1 for Calculus?

1. Differential equations (HL): practice question 2 for Calculus?

1. Differential equations (HL): practice question 3 for Calculus?

1. Differential equations (HL): practice question 4 for Calculus?

1. Differential equations (HL): practice question 5 for Calculus?

1. Differential equations (HL): practice question 6 for Calculus?

1. Differential equations (HL): practice question 7 for Calculus?

1. Differential equations (HL): practice question 8 for Calculus?

1. Differential equations (HL): practice question 9 for Calculus?

1. Differential equations (HL): practice question 10 for Calculus?

An airplane's rate of climb is $c(t)$ meters per minute. At $t=0, 2, 4$ minutes, $c(t)$ is $500, 400, 200$ m/min. Estimate the altitude gained in 4 minutes.

Using the trapezoidal rule with $h=4$ and $y$-values $10, 14, 12, 8$, calculate the area.

If $\int_{1}^{7} f(x) \, dx$ is estimated with $n=3$, what is the step size $h$?

A park's boundary is measured by offsets from a 30m baseline. Offsets are 0m, 10m, 15m, 0m at 10m intervals. Estimate the area.

A worker is painting a wall with a curved top. The height $h$ (m) of the wall at distance $x$ (m) is given by $h(x) = 3 + 0.5x - 0.1x^2$. Use the trapezoidal rule with $n=3$ to estimate the area of the wall from $x=0$ to $x=3$.

Estimate $\int_{1}^{2} \frac{e^x}{x} dx$ using the trapezoidal rule with $n=4$.

The depth of a lake at various distances from the shore is shown below:

$x$ (m) | 0 | 10 | 20 | 30 | 40 $d$ (m) | 0 | 2.5 | 4.8 | 3.2 | 0

Use the trapezoidal rule to estimate the cross-sectional area of the lake.

A set of data points for a function $y=f(x)$ is $(0, 5), (1, k), (2, 11)$. If the trapezoidal rule estimate for $\int_{0}^{2} f(x) dx$ with $n=2$ is 15, find the value of $k$.

The area under $f(x) = \frac{1}{\sqrt{1+x^2}}$ from $x=0$ to $x=1$ is estimated using the trapezoidal rule with $n=2$. Calculate the value.

A sensor measures sound intensity $I(t) = \frac{10}{1+t}$ over 4 seconds. Use the trapezoidal rule with $n=4$ to estimate the total exposure $\int_{0}^{4} I(t) dt$.

The rate of change of a function $g(x) = 5x^2$ at $x = -2$ is:

Calculate the value of $\int_{1}^{e} \frac{1}{x} dx$.

The displacement of a particle is $s(t) = t^3$. Find the average velocity between $t=1$ and $t=3$.

What is the derivative of $f(x) = 7x - 3$?

A particle moves such that its velocity is $v(t) = 10 e^{-t}$. Find the displacement of the particle as $t \to \infty$, assuming $s(0) = 0$.

The cost of producing $x$ items is $C(x) = 200 + 10x + 0.1x^2$. Find the average cost per item $A(x) = \frac{C(x)}{x}$ and find the production level $x$ that minimizes this average cost.

Find the $x$-coordinates of the points where the curve $y = 2x^3 - 3x^2 - 12x + 5$ has horizontal tangents.

A particle's velocity is $v(t) = 4t - t^2$. Find the maximum velocity of the particle.

Calculate the percentage error if the area under $y = x^2$ from $x=0$ to $x=3$ is estimated using the trapezoidal rule with $n=3$.

Find the distance between the two stationary points of $y = x^3 - 3x^2 - 9x + 5$.