2. Intro to Derivatives

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime }\left(a\right)$ exists if and only if $f_{+}^{\prime }\left(a\right)=f_{-}^{\prime }\left(a\right)$.

Compute $f_{+}^{\prime }\left(a\right)$ and $f_{-}^{\prime }\left(a\right)$ at the given point $a$.

$f\left(x\right)=\mid x-2\mid$; $a=2$

The right-sided and left-sided derivatives of a function at a point $a$ are given by $f_{+}^{\prime }\left(a\right)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$ and $f_{-}^{\prime }\left(a\right)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f\left(a\right)}{h}$, respectively, provided these limits exist. The derivative $f^{\prime}\left(a\right)$ exists if and only if $f_{+}^{\prime}\left(a\right)=f_{-}^{\prime}\left(a\right)$.

Compute $f_{+}^{\prime}\left(a\right)$ and $f_{-}^{\prime}\left(a\right)$ at the given point $a$.

$f\left(x\right)=\{\begin{array}{c}4-x^2\text{ if }x\le 1\\ 2x+1\text{ if }x>1\end{array}$; $a=1$

Graph the function $f\left(x\right)=\{\begin{array}{c}x\text{ if }x\le 0\\ x+1\text{ if }x>0\end{array}$.

For x < 0, what is f′(x)?

For x > 0, what is f′(x)?

In Exercises 65 and 66, find the derivative using the definition.

ƒ(t) = 1 .

2t + 1

Graphs

Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).

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Graphs

Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d).

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Consider the function f graphed here. The domain of f is the interval [−4, 6] and its graph is made of line segments joined end to end.

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b. Graph the derivative of f. The graph should show a step function.

Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

f(x) = 4 – x²; f′(−3), f′(0), f′(1)

Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

g(t) = 1/t²; g′(−1), g′(2), g′(√3)

Finding Derivative Functions and Values 

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

f(x) = x + 9/x, x = −3

Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = x³/3

Theory and Examples

In Exercises 51–54,

b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.

y = x⁴/4