Find the linear approximation to f(x)=x2+3x at x=1.

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To approximate a function linearly near a point, we use the tangent line at that point. Start by identifying the function: f(x) = x^2 + 3x. Next, compute the value at the expansion point x0 = 1: f(1) = 1^2 + 3(1) = 1 + 3 = 4.
Then find the derivative: f'(x) = 2x + 3. Evaluate at x0 = 1: f'(1) = 2(1) + 3 = 5.
The linear approximation (tangent line) at x0 is L(x) = f(x0) + f'(x0)(x - x0). Substituting the values gives L(x) = 4 + 5(x - 1).
This expression can be left in point-slope form, i.e., L(x) = 5(x - 1) + 4. If you expand, you obtain L(x) = 5x - 1.
Since the provided answer choices are not available here, we verify that the line 4 + 5(x - 1) correctly represents the linear approximation at x = 1, matching the standard tangent-line formula. The key steps are evaluating the function and its derivative at the expansion point, then forming the tangent line using L(x) = f(1) + f'(1)(x - 1).

DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 5 - L'Hopital's Rule - PreClass

Find the linear approximation to f(x)=x2+3x at x=1.

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DIFFERENTIAL CALCULUS (MTH_251_020_W2025) Unit 5 - L'Hopital's Rule - PreClass

Find the linear approximation to f(x)=x2+3x at x=1.

View Explanation

Log in for full answers

Similar Questions

at 的线性近似值为： 提示：at 的线性近似值为： 。

The linear approximation of 𝑓 ( 𝑥 ) = 5 𝑥 − 4 2 𝑥 2 + 1 at 𝑥 = − 3 is: Hint: The linear approximation of 𝑦 = 𝑓 ( 𝑥 ) at 𝑥 = 𝑥 0 is: 𝐿 ( 𝑥 ) = 𝑓 ′ ( 𝑥 0 ) ( 𝑥 − 𝑥 0 ) + 𝑓 ( 𝑥 0 ) .

This is a continuation of the previous question. What is the Linear Approximation of 𝑓 ( 𝑥 ) = tan ⁡ 𝑥 at 𝑥 = 𝜋 4 ?

Suppose you want to use a Linear Approximation to approximate 1 25.2 without a calculator. Which of the following functions and 𝑥 -values would be best to use?

Consider the Linear Approximation of the function 𝑓 ( 𝑥 ) at 𝑥 = 𝑎 . Which of the following is the best description of what the Linear Approximation does?

Suppose f ( x , t ) = e − 2 t sin ⁡ ( x + 3 t ) . Which of the following is a good approximation of the value of f ( 2.02 , − 0.03 ) ? (Hint: first find the differential d f at the point ( 2 , 0 ) .)

Question at position 3 By using differentials, an approximation of 1233\sqrt[3]{123} is473754\frac{73}{75}.52125\frac{2}{12}.42325.4\frac{23}{25}.497100.4\frac{97}{100}.41315.4\frac{13}{15}.

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at 的线性近似值为：提示：at 的线性近似值为：。