Finding a derivative at a point
requires one to understand tangent lines, concavity, the second derivative, and optimization, as well as the quotient difference rule which tells us that:
It is important to remember that these limit finds are approximations, and that the closer the x value is (the more it approaches 0 aka 0.000000001) to 0, the more accurate the approximation will be.
Word PracticeMARVEL thought it would be wise to release their new film during the winter instead of the summer when more people go to the movies. They were wrong. With a budget of over $8 million dollars, the film only ended up making $3.5 million back and less and less people are trusting to buy it. Although they will never reach 0 customers, MARVEL wants to know how fast the rate of their declining profits are changing. Using to the equation drawn up by a manager below, estimate f'(3.5) when f(x) = 2x^2 to find out how much money, in hundreds, MARVEL is losing on this film per month.
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Numerical PracticeSolve.
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Using tangent lines
Derivative is actually just a fancy word for "slope." Slope, just like a derivative, is a rate of change. By finding the slope of a tangent line of a point on a graph, one can easily find the derivative of that graphed function. Actually, that's exactly what you're doing.
NOTE: A tangent line is a single line at a specific point on a curve
A secant line is a line that connects two points on a curve
NOTE: A tangent line is a single line at a specific point on a curve
A secant line is a line that connects two points on a curve
Khan also explains this topic very well.
The second derivative
The second derivative is exactly what it sounds like, the derivative of the first derivative.
Notation:
Rules:
• If f'(x) < 0 on an interval, then f(x) is decreasing on that interval • If f'(x) > 0 on an interval, then f(x) is increasing on that interval • If f"(x) < 0 on an interval, then f'(x) is decreasing & f(x) is concave down on that interval • If f"(x) > 0 on an interval, then f'(x) is increasing & f(x) is concave up on that interval |
Concavity
When f(x) is concave up, f"(x) is positive. When f(x) is concave down, f"(x) is negative. Every time the second derivative passes through the x-axis f(x) changes concavity.