⧼exchistory⧽
12 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
Let [math]f(x) = 3x^2 + 4[/math]. Using the definition of the derivative, compute
- [math]f^\prime(1)[/math]
- [math]f^\prime(a)[/math], for an arbitrary real number [math]a[/math].
BBy Bot
Nov 03'24
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[/math]
Write an equation of the line tangent to the graph of the function [math]f^\prime[/math] in Problem Exercise at the point
- [math](1,7)[/math]
- [math](a, f(a))[/math].
BBy Bot
Nov 03'24
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[/math]
If [math]F(x) = \frac3{2x+1}[/math], compute [math]F^\prime(3)[/math] using the definition of the derivative.
BBy Bot
Nov 03'24
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[/math]
Using the definition of the derivative, compute [math]f^\prime(a)[/math] for each of the following functions.
- [math]f(x) = x^3[/math]
- [math]f(x) = x^2 + 3x +5[/math]
- [math]f(x) = 7[/math]
- [math]f(x) = \sqrt{x}, a \gt 0[/math]
- [math]f(x) = x + \frac1{x^2}[/math], [math]x \ne 0[/math]
- [math]f(x) = x^3 + 3x^2 + 3x + 1[/math]
- [math]f(x) = \sqrt{x^2 + 1}[/math]
- [math]f(x) = \frac1{\sqrt{x^2 + 1}}[/math]
- [math]f(x) = x^\frac13[/math].
BBy Bot
Nov 03'24
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[/math]
Using the results of Problem Exercise, find an equation of the line tangent to the graph of [math]f[/math] at the point [math](a, f(a))[/math], where
- [math]f(x) = x^3[/math] and [math]a = 0[/math].
- [math]f(x) = x^2 + 3x + 5[/math] and [math]a=1[/math].
- [math]f(x) = 7[/math] and [math]a[/math] is arbitrary.
- [math]f(x) = x + \frac1{x^2}[/math] and [math]a[/math] is not zero.
BBy Bot
Nov 03'24
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[/math]
- If [math]F(x) = x^2[/math], use the definition of the derivative to find [math]F^\prime(x)[/math].
- Plot the graphs of [math]F[/math] and [math]F^\prime[/math] on the same [math]xy[/math]-plane.
BBy Bot
Nov 03'24
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[/math]
- Show that the function [math]|x|[/math] is not differentiable at [math]0[/math] and interpret this fact geometrically.
- Compute the derivative at [math]-1[/math] and at [math]1[/math] of the function [math]|x|[/math].
BBy Bot
Nov 03'24
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Show that the function [math]\sqrt x[/math] is not differentiable at [math]0[/math]. Draw the graph and interpret the nondifferentiability geometrically.
BBy Bot
Nov 03'24
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[/math]
Using the results of Problem Exercise, find
- [math]\ddxof f (-1)[/math] if [math]f(x) = x^2 + 3x + 5[/math].
- [math]\ddxof f (3)[/math] if [math]f(x) = x^3[/math].
- [math]\ddxof f (b)[/math] if [math]f(x) = x^3 + 3x^2 + 3x + 1[/math].
- [math]\ddxof{\sqrt{x^2 + 1}}[/math].
- [math]\ddxof{(x^2 + 3x + 5)} (a)[/math].
- [math]\ddx \left( x + \frac1{x^2} \right) [/math].
BBy Bot
Nov 03'24
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[/math]
- If [math]y = 2x + 1[/math], find [math]\dydx (a)[/math].
- If [math]s = 16t^2[/math], find [math]\ddxof s (2)[/math].
- If [math]s = 16t^2[/math], find [math]\ddxof s[/math].