CPM Homework Help : CALC Problem 7-65

CPM Homework Banner

Home > CALC > Chapter 7 > Lesson 7.2.1 > Problem 7-65

7-65.

No calculator! Given $f(x) = 3^x − 2$ and $g(x) =\frac { 1 } { 2 }\operatorname{sin}x$ , find the following limits:

$\lim\limits_{ x \rightarrow 1 } f ^ { \prime } ( x )$
Hint (a):
$\frac{d}{dx}a^{x}=\text{ln}x\cdot a^{x}$

$\lim\limits_ { x \rightarrow \pi / 4 } g ^ { \prime \prime } ( x )$
Hint (b):
$\frac{d}{dx}\text{sin}x=\text{cos}x; \frac{d}{dx}\text{cos}x=-\text{sin}x$

$\lim\limits_{ x \rightarrow \pi } f ( g ( x ) )$
Hint (c):
$\lim\limits_{x\to\pi}(3^{0.5\sin(x)}-2)$

$\lim\limits_{ x\rightarrow\pi / 2 }f ( g ^ { \prime } ( x ) )$
Hint (d):
Be mindful about positive and negative values.

$\lim\limits_{ h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$
Hint (e):
This is Hana's definition of the derivative.

$\lim\limits_ { x \rightarrow \frac { \pi } { 4 } } \frac { g ( x ) - \frac { \sqrt { 2 } } { 4 } } { x - \frac { \pi } { 4 } }$
Hint 1(f):
This is Ana's definition of the derivative.
Hint 2(f):
$g'\left ( \frac{\pi }{4} \right )={\_\_\_\_\_}$

Compute without a calculator

© 2022 CPM Educational Program. All rights reserved.