The Fundamental Theorem of Calculus binds together the concepts of differentiation and integration. The process of deriving using this theorem involves integrating a function, followed by differentiating the outcome. Basically, what this theorem signifies is that the original function can be retrieved by integrating and then differentiating it, or the other way around.
| Topic | Problem | Solution |
|---|---|---|
| None | The surface area of a human (in square meters) ha… | Given the function for the surface area of a human, \(A = 0.024265h^{0.3964}m^{0.5378}\), where \(h… |
| None | The radius of a circle is increasing at a rate of… | The problem involves a circle with a radius that is increasing at a rate of 10 centimeters per minu… |
| None | For the following demand equation, differentiate … | First, we clear the fraction in the given equation. We multiply both sides by \(x+p\) to get \(6xp … |
| None | Pierce Manufacturing determines that the daily re… | The current daily revenue is \(\$ 3800\). |
| None | Certain chemotherapy dosages depend on a patient'… | Given that Kim's height is a constant 161 cm, and she is losing weight at a rate of 5 kg per month,… |
| None | Suppose $f(x)$ is a mystery function, where $f^{\… | Suppose $f(x)$ is a mystery function, where $f^{\prime \prime}(x)=8$ and $f^{\prime}(4)=6$. The que… |
| None | In a trend that scientists attribute, at least in… | The problem is asking for the rate of change of the area of the ice cap. This is a problem of relat… |
| None | Find the rate of change of total revenue, cost, a… | Given the revenue function \(R(x) = 2x\) and the cost function \(C(x) = 0.01x^2 + 0.2x + 5\), we ar… |
| None | Assume that $x=x(t)$ and $y=y(t)$. Let $y=x^{2}+5… | We are given that \(y=x^{2}+5\) and \(\frac{d x}{d t}=4\) when \(x=3\). We are asked to find \(\fra… |
| None | The table below gives the height above the ground… | The table below gives the height above the ground, $h$, of a passenger traveling on the Vegas High … |
| None | The area of a healing wound is given by $A=\pi r^… | We are given that the area of a healing wound is given by the formula \(A=\pi r^{2}\), where \(r\) … |
| None | If $f^{\prime}(x)=15 x^{2}-4$ and $f(x)$ passes t… | Given that the derivative of the function, \(f'(x) = 15x^2 - 4\), we can find the original function… |
| None | Find the gradient of $f(x, y, z)=\left(x^{2}+y^{2… | Given the function \(f(x, y, z)=(x^{2}+y^{2}+z^{2})^{-1 / 2}+\ln (x y z)\), we need to find the gra… |
| None | When a bactericide is added to a nutrient broth i… | The size of the bacteria population at time $t$ (hours) is given by the function $b=6^{7}+6^{5} t-6… |
| None | The cost function for a certain commodity is \[ C… | Given the cost function \(C(q)=88+0.19 q-0.007 q^{2}+0.0008 q^{3}\) |
| None | The weekly marginal cost of producing $x$ pairs o… | The weekly marginal cost of producing \(x\) pairs of tennis shoes is given by the function \(C^\pri… |
| None | The marginal average cost of producing $x$ digita… | First, we need to find the average cost function. We know that the derivative of the average cost f… |
| None | Consider the following. \[ 4 x^{5}+y^{3}=9 x \] (… | Differentiate both sides of the equation \(4x^5 + y^3 = 9x\) with respect to x. The derivative of a… |
| None | Suppose that the price $p$, in dollars, and the n… | Differentiate the given equation with respect to time $t$ to get an equation involving $\frac{dp}{d… |
| None | Find the rate of change of total revenue, cost, a… | Given the revenue function \(R(x) = 55x - 0.5x^2\), the cost function \(C(x) = 2x + 20\), and \(dx/… |
| None | A circle is growing, its radius increasing by $4 … | The area of a circle is given by the formula \(A = \pi r^2\). |
| None | A particular computing company finds that its wee… | Let's denote the number of laptops produced and sold weekly as \(x\) and the weekly profit as \(P(x… |
| None | The demand, $D$, for a new rollerball pen is give… | We are given the demand function, \(D = 0.009p^{3} - 0.5p^{2} + 180p\), where \(p\) is the price in… |
| None | At time $t$, the position of a body moving along … | The position of the body is given by \(s=-t^{3}+12 t^{2}-45 t m\). |
| None | The volume of a cube decreases at a rate of $0.8 … | We are given that the volume of a cube decreases at a rate of \(-0.8 \, \text{ft}^{3} / \text{min}\… |
| None | Use logarithmic differentiation to evaluate $y^{\… | First, we take the natural logarithm of both sides of the equation to simplify the differentiation … |
| None | A cylinder begins with a diameter of 28 yards and… | We are given a cylinder with a diameter of 28 yards and a height of 22 yards. The diameter is incre… |
| None | Find $\nabla f$ at the given point. \[ f(x, y, z)… | Find the partial derivatives of the function: \(f(x, y, z) = x^{3} + y^{3} - 4z^{2} + z \ln x\) |
| None | Find how fast the circumference of a circl is gro… | Given the rate at which the radius is growing: \(\frac{dr}{dt} = 7 \frac{\text{cm}}{\text{s}}\) |
| None | A particle moves according to the equation \( x=1… | \(v(t) = \frac{dx}{dt} = 144t^{8}\) |
| None | The position of a particle moving along \( x \) a… | \(x(t) = t^3 + 8t^2 + 35\) |