Biggest slope of a curve, calculus 1 optimization  Summary and Q&A
TL;DR
Learn how to find the points on a curve where the tangent line has the biggest slope.
Key Insights
 βΊοΈ Differentiating the equation once gives the derivative, which represents the slopes at each xvalue.
 β To find the maximum slope, differentiate the derivative equation again.
 π± Factoring the equation helps identify the critical numbers, which are the xvalues where the slope changes.
 π The first derivative test is used to determine if the critical numbers are local maxima or minima.
 π₯ The local maxima represent the points on the curve with the largest slope.
 π The local maxima can be found by plugging the critical numbers back into the original equation.
 π₯οΈ The largest slope was found to be 240.
Transcript
okay thanks PPR we're gonna find out which points on the curve why it's equal to one plus 40 X to a third power minus three x plus v power that's the tangent line have the biggest slope this question right here it's actually kind of no confusing for some students so be sure you pay close attention to this first of all we are looking for the biggest... Read More
Questions & Answers
Q: How do you find the biggest slope of the tangent line?
To find the biggest slope, differentiate the equation and then differentiate it again. Use the critical numbers obtained from factoring and apply the first derivative test to find the local maxima.
Q: How many critical numbers can be obtained from the equation?
From factoring the equation, three critical numbers can be obtained: 0, 2, and 2.
Q: Why do we need to consider the local maxima for finding the biggest slope?
We need to consider the local maxima because they represent the points on the curve where the tangent line has the largest slope.
Q: How do we calculate the slope at the critical numbers?
Plug in the critical numbers into the derivative equation to calculate the slope at those specific points.
Summary & Key Takeaways

To find the biggest slope of the tangent line, differentiate the given equation once.

Differentiate the equation again to find the maximum slope.

Factor the equation to find the critical numbers and use the first derivative test to determine the local maxima.