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Chapter 2: Q. 95 (page 186)

Use the mathematical definition of a tangent line and the point-slope form of a line to show that if f is differentiable at x=c, then the tangent line to f at x=cis given by the equation y=f'(c)(x鈭�c)+f(c).

Short Answer

Expert verified

Ans:

y-f(c)=f'(c)[x-c]y=f'(c)[x-c]+f(c)

Step by step solution

01

Step 1. Given information:

Consider a function: f(x)

Here the objective is to show that the equation of the tangent line at any point x=c is

y=f(c)[x-c]+f(c)

02

Step 2. Showing that if f is differentiable at x=c:

Let the derivative of the function is f'(x).

At point x=cthe slope of the tangent line is: f'(x)and the value of the function is: f(c) Therefore equation of the tangent line at (c,f(c))with slope f'(x)is:

y-f(c)=f'(c)[x-c]y=f'(c)[x-c]+f(c)

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Most popular questions from this chapter

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x+1x2+1

For each function fgraphed in Exercises 65-68, determine the values of xat which ffails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Differentiate f(x)=3x+x2in three ways. When you have completed all three parts, show that your three answers are the same:

(a) with the chain rule

(b) with the product rule but not the chain rule

(c) without the chain or product rules.

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

26.f(x)=1x2,x=2

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

a)f(x)=(3x+1)4b)f(x)=(1x+1x2)2c)f(x)=(x+1)2xd)f(x)=(x+1)2x
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