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Chapter 2: Q. 1TF (page 168)

Taking the limit: We have seen that if f is a smooth function, then f'(c)鈮�f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh鈫�0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

Short Answer

Expert verified

The exact slope of the tangent is 8.

Step by step solution

01

Step 1.Given Information

Given equation isf(x)=x2atx=4.

02

Step 2.Find the derivative 

Formula for derivative f'(c)=limh鈫�0f(c+h)-f(c)h

Here x鈫�c;x鈫�4;c鈫�4

f'(4)=limh鈫�0f(4+h)-f(4)h=limh鈫�04+h2-42h=limh鈫�042+h2+8h-42h=limh鈫�0h2+8hh=limh鈫�0h(h+8)h=(0+8)=8.

03

Step 3.The solution

The exact slope of the tangent is 8.Since the derivative of the equation at point gives the slope.

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Most popular questions from this chapter

Use the definition of the derivative to find ffor each function fin Exercises 39-54.

f(x)=x2-1x2-x-2

Prove that if f is any cubic polynomial function f(x)=ax3+bx2+cx+dthen the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

a=f"'(0)6;b=f"(0)2;c=f'(0);d=f(0)

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)=x13-2x-1

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=3x5-2x2+4;f(0)=1

Prove that if f is a quadratic polynomial function f(x)=ax2+bx+cthen the coefficient of f are completely determined by the values of f(x) and its derivatives at x=0 as follows

a=f"(0)2;b=f'(0);c=f(0)
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