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With what initial velocity must an object be projected vertically upward from the ground to reach a height of 600 feet?

(a) $9 x \sqrt{3 x^{2}+9}$(b) $\frac{-3 x\left(x^{2}-1\right)^{2} \sqrt{3 x^{2}+9}}{x^{2}+1}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 7

Marginal Functions and Rates of Change

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

05:46

With what initial velocity…

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01:37

With what velocity must an…

04:08

Assume the acceleration du…

01:25

04:59

00:56

An object is dropped from …

07:04

A ball is thrown verticall…

01:47

A projectile is launched a…

01:58

Use the formula for the he…

01:17

What is the maximum height…

04:48

$\bullet$ $\bullet$ $\bull…

02:52

No. So for this one, you know what t one at the time that the balls reaches the top. Washington Monday. Okay, that case, you know that Lee of t one minus zero physical two. Now they have to be one zero. And today, these negatives 32 we're seconds squared to my demands. Yeah, this 32 two. No. You also know because 5 15 it is the height of the Washington Mother is equal to They are too t squared. It's a t one squared course. The velocity of time. 21 Close the position at the initial time. Zero. Well, that cereal, that's zero. So that means that 5 50 is equal to 32 over to you. T one squared, which happens to be 16 t one squared. That tells us that t one squared 5 50 or her 16. Yeah. One is the square. 5 15 over 16 for bring 5 50 over four, which implies a commission velocity. No, I was able to 32 times 15 four. Cheers. No, you 15. Okay. Makes sense

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