Finding the Derivative using the Fundamental Theorem of Calculus Introduction,Examples with Step-by-Step Explanations

The problems about Finding the Derivative using the Fundamental Theorem of Calculus

Topic	Problem	Solution
None	The surface area of a human (in square meters) ha…	Given the function for the surface area of a human, $A = 0.024265 h^{0.3964} m^{0.5378}$ , 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 $f^{''} (x) = 8$ and $f^{'} (4) = 6$ . 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 $R (x) = 2 x$ and the cost function $C (x) = 0.01 x^{2} + 0.2 x + 5$ , 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 $A = π r^{2}$ , where $r$ …
None	If $f^{'} (x) = 15 x^{2} - 4$ and $f (x)$ passes t…	Given that the derivative of the function, $f^{'} (x) = 15 x^{2} - 4$ , 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 $4 x^{5} + y^{3} = 9 x$ 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 $R (x) = 55 x - 0.5 x^{2}$ , the cost function $C (x) = 2 x + 20$ , and \(dx/…
None	A circle is growing, its radius increasing by $4 …	The area of a circle is given by the formula $A = π r^{2}$ .
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, $D = 0.009 p^{3} - 0.5 p^{2} + 180 p$ , 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: $f (x, y, z) = x^{3} + y^{3} - 4 z^{2} + z \ln x$
None	Find how fast the circumference of a circl is gro…	Given the rate at which the radius is growing: $\frac{d r}{d t} = 7 \frac{cm}{s}$
None	A particle moves according to the equation \( x=1…	$v (t) = \frac{d x}{d t} = 144 t^{8}$
None	The position of a particle moving along $x$ a…	$x (t) = t^{3} + 8 t^{2} + 35$