Position Function From Velocity

Given the circle defined by $$x^2+y^2=25$$, use implicit differentiation to solve for $$\frac{dy}{dx}$$ at the point $$(3,4)$$.

Differentiability and Continuity

Alternating Series Test Convergence

Volume Of A Solid Of Revolution

Tangent Line Approximation

A particle’s position is given by $$s(t)=-t^2+4*t+1$$. By applying the Mean Value Theorem on the interval $$[0,5]$$, at what time does the instantaneous velocity equal the average velocity?

Constant of Integration in SIPPY Method

Growth Rate In Logistic Differential Equations

A conical tank has water whose height $$h$$ (in meters) and radius $$r$$ are related by $$r=0.5*h$$. The volume of water is given by $$V=\frac{\pi}{3}r^2h$$. Using the relation between $$r$$ and $$h$$, show that $$V=\frac{\pi}{12}h^3$$. Then, if at a certain moment the water height is $$h=3\,m$$ and the rate of change of the height is $$\frac{dh}{dt}=0.5\,m/s$$ (as indicated by the graph), calculate the rate at which the volume is changing, $$\frac{dV}{dt}$$.

General Form of an Alternating Series

Comparing Riemann Sum Accuracy

Limit of a Rational Function

Chain Rule with a Logarithmic Function

First Derivative and Function Behavior

Integral of a Vector-Valued Function

A table extracted from the graph provides the following values for a function $$H(t)$$: Years (t): 2, 3, 5, 7, 10 H(t): 1.5, 2, 6, 11, 15 Using the trapezoidal rule, estimate the definite integral of $$H(t)$$ with respect to t over the interval from 2 to 10.

Definition of Continuity at a Point

Function Value and Average Rate of Change

Area of a Square Cross Section