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Questions and Answers
If f(x) = -3x(x + 2)², then what is the slope of the tangent line to the graph when x = -1?
Find lim as h approaches 0 of (x + h - x)/h
If g'(1) = -3, then which of the following could be the equation for g(x)?
Which of the following statements is/are true about f'(x) for the polynomial function f(x)?
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Which of the following would represent f'(x) if f(x) = (x² + 2x)/(3x + 1)?
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If g'(x) = -3x(x + 2)², then the graph of g(x) has a relative maximum at what value(s) of x?
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Show algebraically that f'(x) = 1 + 3x sin(x) for f(x) = 2x - 3 cos(x).
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Will the slope of the normal line drawn to the graph of f at x = 4 be positive or negative?
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Study Notes
Derivatives and Tangent Lines
- The derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.
- The slope of the tangent line to the graph of f(x) = -3x(x+2)^2 at x = -1 is -3.
Limits and Derivatives
- The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches 0:
- lim h->0 [f(x+h) - f(x)] / h
- The limit as h approaches 0 of (x+h - x) / h is x.
Derivatives and Function Properties
- If g'(1) = -3, then the function g(x) could be represented by equations that have a derivative equal to -3 when x = 1.
- The derivative of a polynomial function f(x) is negative when f(x) is decreasing.
- The derivative of a polynomial function f(x) changes from negative to positive when f(x) has a relative minimum.
- The derivative of a polynomial function f(x) is equal to zero at the points where f(x) has a horizontal tangent line.
Finding Derivatives
- The derivative of f(x) = (x^2 + 2x) / x is (3x + 1) / 2x.
The Derivative and the Shape of a Function
- If the derivative of a function f(x) is positive, then the function is increasing.
- If the derivative of a function f(x) is negative, then the function is decreasing.
- If the derivative of a function f(x) is zero, then the function has a horizontal tangent line, which may be a relative maximum, minimum, or neither.
- The slope of the normal line at a point is the negative reciprocal of the slope of the tangent line at that point.
Finding Relative Extrema (Maxima and Minima)
- If the derivative of a function f(x) changes from positive to negative at a point x = a, then f(x) has a relative maximum at x = a.
- If the derivative of a function f(x) changes from negative to positive at a point x = a, then f(x) has a relative minimum at x = a.
- The graph of f(x) = 2x - 3cosx has a relative maximum at x = 1.895 and a relative minimum at x = 4.937.
Determining Intervals of Increase and Decrease
- The function f(x) is increasing on the intervals (0, 1.895) and (4.937, 2π).
- The function f(x) is decreasing on the interval (1.895, 4.937).
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Description
This quiz covers key concepts related to derivatives and tangent lines in calculus. Understand how to find the derivative at specific points, interpret the slope of tangent lines, and apply limits in derivative calculations. Test your knowledge on how these concepts apply to polynomial functions.
