/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Find the derivative by the limit... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 15

Find the derivative by the limit process. \(f(x)=2 x^{2}+x-1\)

Short Answer

Expert verified
The derivative of the function \(f(x)=2x^{2}+x-1\) based on the limit process is \(f'(x)=4x+1\)

Step by step solution

01

Define the Problem

We need to find the derivative \(f'(x)\) of the function \(f(x)=2x^{2}+x-1\) using the limit process. The formula for the limit process or definition of the derivative is \[f'(x)=\lim_{h\to 0}\frac{f(x+h) - f(x)}{h}\]
02

Substitute the function into the Formula

We start by substituting \(f(x)=2x^{2}+x-1\) into the limit formula. This means we need to calculate \(f(x+h)\) and then subtract \(f(x)\): \[f'(x)=\lim_{h\to0}\frac{(2(x+h)^{2}+(x+h)-1) - (2x^{2}+x-1)}{h}\]
03

Simplify the Expression

Next, simplify the expression in the numerator. This involves expanding the term \((x+h)^{2}\), simplifying the resulting expression and cancelling out any common terms: \[=\lim_{h\to0}\frac{(2x^{2}+4xh+2h^{2}+x+h-1)-(2x^{2}+x-1)}{h} =\lim_{h\to0}\frac{4xh+2h^{2}+h}{h}\]
04

Further Simplify the Expression

Now factor out an \(h\) from the numerator and simplify the expression: \[f'(x) = \lim_{h \to 0}(4x+2h+1)\]
05

Apply Limit to the Expression

Lastly, apply the limit to the simplified expression: As \(h\) approaches 0, \(2h\) will also approach 0, which leaves us with: \[f'(x)=4x+1\]

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Most popular questions from this chapter

Linear and Quadratic Approximations In Exercises 33 and 34, use a computer algebra system to find the linear approximation $$P_{1}(x)=f(a)+f^{\prime}(a)(x-a)$$ and the quadratic approximation $$P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}$$ to the function \(f\) at \(x=a\). Sketch the graph of the function and its linear and quadratic approximations. $$ f(x)=\arccos x, \quad a=0 $$

Prove that \(\arcsin x=\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\)

Determine the point(s) at which the graph of \(y^{4}=y^{2}-x^{2}\) has a horizontal tangent.

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=4-x^{2}-\ln \left(\frac{1}{2} x+1\right)} \quad \frac{\text { Point }}{\left(0,4\right)}\)

In Exercises 43 and 44, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x\).
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