Explanation Given: Function f ( x ) = x 3 e x . Formula Used: Equation of line using slope-point form is given as ( y − y 1 ) = m ( x − x 1 ) , where ( x 1 , y 1 ) is point of line and m is the slope of the line. Calculation: Function is given as f ( x ) = x 3 e x . Let us find the point on graph where tangent is drawn Put x = 2 in f ( x ) , we get f ( 2 ) = ( 2 ) 3 e 2 f ( 2 ) = 8 e 2 . Also, slope of the line is calculated by taking the derivative of the function f ′ ( x ) = d d x ( x 3 e x ) Using product rule of derivative, we have f ′ ( x ) = d d x ( x 3 ) × ( e x ) + x 3 × d d x ( e x ) f ′ ( x ) = ( 3 x 2 ) ( e x ) + x 3 ( e x ) f ′ ( x ) = 3 x 2 e x + x 3 e x

6th Edition

ISBN: 9781119399353

Author: DA

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3.4, Problem 37P

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To find the equation of tangent line to the graph of f(x)=x3ex at the point at which x=2.

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Chapter 3.4, Problem 36P

Chapter 3.4, Problem 38P

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Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter | (for "incorrect") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) ☐ 1. For all n > 1, seriesΣ In(n) In(n) converges. 2, 1, arctan(n) the series arctan(n) n³ ☐ 4. For all n > 1, 123 converges. 1 n ln(n) series In(n) diverges. 2n . and the seriesΣconverges, so by the Comparison Test, 2, 3, and the series converges, so by the Comparison Test, the series-3 1 converges. ☐ 6. For all n > 2, In(n) >, and the series Σ converges, so by the Comparison Test, the seriesΣ In(n) converges.

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To find the equation of tangent line to the graph of f ( x ) = x 3 e x at the point at which x = 2 .

Solution Summary: The author explains how to find the equation of tangent line using slope-point form.

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To find the equation of tangent line to the graph of f(x)=x3ex at the point at which x=2.

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Chapter 3 Solutions