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Quiz: Derivative by definition



Practice Problems



Problem 1) Find the average rate of change of $$ f(x) = \sqrt{4x +1 } $$ over the interval [0,2]

Answer
problem 1


Problem 2) Find the average rate of change of $$ f(x) = e^x$$ over the interval [0,2]

Answer
problem2


Problem 3) Find the slope of the secant line between $$\text{x = } \frac{\pi}{6} \text{ and x = } \frac{\pi}{3} \text{ on } f(x) = cos(x) $$

Answer
problem 3


Problem 4) Find the slope of the secant line between $$ f(x) = ln(x) [1,4] $$

Answer
problem 3


Problem 5) Use the definition $$ \lim\limits_{h\rightarrow 0} \frac{(f_{0} + h)-f(x_{0})}{h} $$ to find the slope of the tangent line (derivative) of the given function at the indicated point.

$$ f(x) = x^2 + 3x \text{ at x = } 1 $$

Answer
problem 3
$$ f(x) = 3 -5x^2 \text{ at x = } 1 $$

Answer
part a


$$ f(x) = -\frac{1}{x^2} \text{ at x = } 2 $$

Answer
part b


$$ f(x) = 2\sqrt{x} \text{ at x = } 1 $$

Answer
$f(x)=2\sqrt{x} \text{ at x = }1$ $\lim\limits_{h \rightarrow 0}\frac{f(x_{0}+h)-f(x_{0})}{h} \\ \lim\limits_{h \rightarrow 0}\frac{2\sqrt{(1+h)}-2\sqrt{(1)}}{h}$ $\lim\limits_{h \rightarrow 0}\frac{2\sqrt{(1+h)}-2}{h}$ $\lim\limits_{h \rightarrow 0}\frac{2\sqrt{(1+h)}-2}{h}\Bigg(\frac{2\sqrt{(1+h)}+2}{2\sqrt{(1+h)}+2}\Bigg)$ $\lim\limits_{h \rightarrow 0}\frac{(2\sqrt{(1+h)}-2)(2\sqrt{(1+h)}+2)}{h(2\sqrt{(1+h)}+2)}$ $\lim\limits_{h \rightarrow 0}\frac{4(1+h)-4}{h(2\sqrt{(1+h)}+2)}$ $\lim\limits_{h \rightarrow 0}\frac{4h}{h(2\sqrt{(1+h)}+2)} \\ \lim\limits_{h \rightarrow 0}\frac{4}{(2\sqrt{(1+h)}+2)} = \\ \frac{4}{(2\sqrt{(1+0)}+2)} = \frac{4}{(2+2)} = \frac{4}{(4)} = 1$