Question
(a) Show that if $f$ and $g$ are functions for which$$f^{\prime}(x)=g(x) \text { and } g^{\prime}(x)=-f(x)$$for all $x,$ then $f^{2}(x)+g^{2}(x)$ is a constant.(b) Give an example of functions $f$ and $g$ with this property.
Step 1
We want to show that $f^2(x) + g^2(x)$ is a constant. Show more…
Show all steps
Your feedback will help us improve your experience
Sriram Soundarrajan and 50 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
(a) Show that if $f$ and $g$ are functions for which $$ f^{\prime}(x)=g(x) \text { and } g^{\prime}(x)=f(x) $$ for all $x,$ then $f^{2}(x)-g^{2}(x)$ is a constant. (b) Show that the function $f(x)=\frac{1}{2}\left(e^{x}+e^{-x}\right)$ and the function $g(x)=\frac{1}{2}\left(e^{x}-e^{-x}\right)$ have this property.
THE DERIVATIVE IN GRAPHING AND APPLICATIONS
Rolle’s Theorem; Mean-Value Theorem
a) Show that if $f(x)$ and $g(x)$ are functions such that $f(x)$ is $o(g(x))$ and $c$ is a constant, then $c f(x)$ is $o(g(x))$ where $(c f)(x)=c f(x)$ b) Show that if $f_{1}(x), f_{2}(x),$ and $g(x)$ are functions such that $f_{1}(x)$ is $o(g(x))$ and $f_{2}(x)$ is $o(g(x)),$ then $\left(f_{1}+\right.$ $f_{2} )(x)$ is $o(g(x)),$ where $\left(f_{1}+f_{2}\right)(x)=f_{1}(x)+f_{2}(x)$
Algorithms
The Growth of Functions
Prove that if $f^{\prime}(x)=g^{\prime}(x)$ for all $x$ in $(a, b),$ then there is a constant $C$ such that $f(x)=g(x)+C$ on $(a, b) .$ [Hint: Apply the Constant Function Theorem to $h(x)=f(x)-g(x) .]$
Short-Cuts to Differentiation
Theorems About Differentiable Functions
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD