Chapter 3: Problem 4
In Exercises \(1-6,\) find \(d y / d x\). $$y=x^{2}+x+1$$
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Multiple Choice Find \(y^{\prime \prime}\) if \(y=x \sin x\) (A) \(-x \sin x\) (B) \(x \cos x+\sin x\) (C) \(-x \sin x+2 \cos x\) (D) \(x \sin x\) (E) \(-\sin x+\cos x\)
Finding \(f\) from \(f^{\prime}\) Let $$f^{\prime}(x)=3 x^{2}$$ (a) Compute the derivatives of \(g(x)=x^{3}, h(x)=x^{3}-2,\) and \(t(x)=x^{3}+3 .\) (b) Graph the numerical derivatives of \(g, h,\) and \(t\) (c) Describe a family of functions, \(f(x),\) that have the property that \(f^{\prime}(x)=3 x^{2}\) . (d) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=0 ?\) If so, what is it? (e) Is there a function \(f\) such that \(f^{\prime}(x)=3 x^{2}\) and \(f(0)=3 ?\) If so, what is it?
Inflating a Balloon The volume \(V=(4 / 3) \pi r^{3}\) of a spherical balloon changes with the radius (a) At what rate does the volume change with respect to the radius when \(r=2 \mathrm{ft} ?\) (b) By approximately how much does the volume increase when the radius changes from 2 to 2.2 \(\mathrm{ft}\) ?
Particle Motion The position of a body at time \(t\) sec is \(s=t^{3}-6 t^{2}+9 t \mathrm{m} .\) Find the body's acceleration each time the velocity is zero.
Exploration Let \(y_{1}=a^{x}, y_{2}=\mathrm{NDER} y_{1}, y_{3}=y_{2} / y_{1},\) and \(y_{4}=e^{y_{3}}\) (a) Describe the graph of \(y_{4}\) for \(a=2,3,4,5 .\) Generalize your description to an arbitrary \(a>1\) (b) Describe the graph of \(y_{3}\) for \(a=2,3,4,\) 5. Compare a table of values for \(y_{3}\) for \(a=2,3,4,5\) with \(\ln a\) . Generalize your description to an arbitrary \(a>1\) (c) Explain how parts (a) and (b) support the statement \(\frac{d}{d x} a^{x}=a^{x} \quad\) if and only if \(\quad a=e\) (d) Show algebraically that \(y_{1}=y_{2}\) if and only if \(a=e\) .
What do you think about this solution?
We value your feedback to improve our textbook solutions.
