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# Week 7 Problem Set C.
## Reading.
Thomas Chapter 3.1, 3.2, 3.3, 3.4, 3.5.
## Some mixed practice.
### Slopes and Tangents.
1. For this problem, consider the curve $y=x^{3}-4x+1$.
1. **Normal to a curve.** Find an equation for the line **perpendicular** to the tangent line of this curve at the point $(2,1)$.
2. **Smallest slope.** What is the smallest slope on this curve? At what point on the curve does the curve have this slope? (Hint: If you have a quadratic function, you find the smallest value by putting the quadratic in vertex-form)
3. **Tangents having specified slope.** Find equations for the tangents to the curve at the points where the slope of the curve is $8$.
2. For this problem, consider the curve $y=x^{3}-3x-2$.
1. **Horizontal tangent.** Find equations for the horizontal tangents (zero slope) to the curve $y=x^{3}-3x-2$. What are the equations of the **normal lines** (lines that are perpendicular to the tangent lines) to these points of tangency?
2. **Smallest slope.** What is the smallest slope on this curve? At what point on the curve does the curve have this slope? Find an equation for the line that is perpendicular to the curve's tangent at this point.
3. Consider **Newton's serpentine** $\displaystyle y= \frac{4x}{x^{2}+1}$. Find the tangent line equations at the origin $(0,0)$, and at the point $(1,2)$.
3. **Quadratic tangent to identity function.** The curve $y=ax^{2}+bx + c$ passes through the point $(1,2)$, and is tangent to the line $y=x$ at $(0,0)$. Find $a,b,c$. **(There was a typo that has been corrected.)**
4. **Quadratics having a common tangent.** The curves $y=x^{2}+ax+b$ and $y=c-x^{2}$ have a common tangent line at the point $(1,0)$. Find $a,b,c$.
5. Find all points $(x,y)$ on the graph of $f(x)=3x^{2}-4x$ with tangent lines parallel to the line $y=8x+5$
6. Find all points $(x,y)$ on the graph of $y=x / (x-2)$ with tangent lines perpendicular to the line $y=2x+3$.
7. Below shows graph of $y,y',$ and $y''$. Which one is which? Can you briefly explain?![[1 teaching/smc-fall-2023-math-7/week-7/---files/Pasted image 20231013113718.png]]
8. Below shows graphs of $y,y'$ and $y''$. Which one is which? Can you briefly explain?![[1 teaching/smc-fall-2023-math-7/week-7/---files/Pasted image 20231013113751.png]]
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