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 |
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None | The surface area of a human (in square meters) ha… | Given the function for the surface area of a human, |
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 |
None | Pierce Manufacturing determines that the daily re… | The current daily revenue is |
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 |
Suppose |
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 |
None |
Assume that |
We are given that |
None | The table below gives the height above the ground… | The table below gives the height above the ground, |
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 |
None |
If |
Given that the derivative of the function, |
None | Find the gradient of $f(x, y, z)=\left(x^{2}+y^{2… | Given the function |
None | When a bactericide is added to a nutrient broth i… | The size of the bacteria population at time |
None | The cost function for a certain commodity is \[ C… | Given the cost function |
None |
The weekly marginal cost of producing |
The weekly marginal cost of producing |
None |
The marginal average cost of producing |
First, we need to find the average cost function. We know that the derivative of the average cost f… |
None |
Consider the following.
|
Differentiate both sides of the equation |
None |
Suppose that the price |
Differentiate the given equation with respect to time |
None | Find the rate of change of total revenue, cost, a… | Given the revenue function |
None | A circle is growing, its radius increasing by $4 … | The area of a circle is given by the formula |
None | A particular computing company finds that its wee… | Let's denote the number of laptops produced and sold weekly as |
None |
The demand, |
We are given the demand function, |
None |
At time |
The position of the body is given by |
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 |
Find the partial derivatives of the function: |
None | Find how fast the circumference of a circl is gro… | Given the rate at which the radius is growing: |
None | A particle moves according to the equation \( x=1… | |
None |
The position of a particle moving along |