CPM Homework Help : PC Problem 10-29

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10-29.

Approximate the slope of the tangent line (instantaneous rate of change) for the function $f(x)=5^x-x^3$ at $x=1$ .

Hint:

$\lim\limits_ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$

Step 1:

$\frac{5^{x+h}-(x+h)^3-(5^x-x^3)}{h}$

Step 2:

$\frac{(5^{x+h}-5^x)-(x^3+3x^2h+3xh^2+h^3)+x^3}{h}$

Step 3:

$\frac{5^x(5^h-1)-3x^2h-3xh^2-h^2}{h}$

Step 4:

Let $x=1$ and $h=0.001$ .

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