# Calculus

Tangent Lines

1. What is the slope of the line tangent to $f(x)=x^2+3x$ at $x=3$?

2. Negan's favorite function is $N(x)=\sqrt{x^6-x^5}$. What is the slope of the tangent line to $N(x)$ at $x=2$?

3. What is the slope of the tangent line to the graph $y=x^2$ at the point $(3,9)$?

4. What is the equation of the line tangent to the graph of $y=\sin^2x$ at $x=\frac{\pi}{4}$?

5. At what positive $x$ value on the curve $f(x)=2x^3-9x^2-12x+10$ is the tangent to the curve at that point parallel to the line $y=12x+7$?

6. The equation of the line tangent to $h(x)=x^3+2x$ when $x=1$ is $y=Ax+B$ where $A$ and $B$ are real numbers. Find $A^2 +B^2$.

7. Find the slope of the tangent to the graph of the equation $y=\log_2x$ at the point $x=-1$.

8. Find the slope of the function $3x^2y-4x^3=1$ at $x=2$.

9. At what value of $t$ does the curve $x=t^2-t, y=t^2+t$ have a vertical tangent?

10. Find the slope of the tangent to the curve with equations $x=t+t^2, y=t+e^t$ at the point $(0,1)$

11. Let $f(x)$ be defined on the interval $[0,\infty)$. What is the equation of the line tangent to the function $f(x)=x^3+2x^2-x-2$ at $y=0$?

12. Use the linear approximation of $f(x)=\sqrt{x+1}$ near $x=0$ to approximate $\sqrt{1.02}$.

13. If $f(x)=\sqrt{x}$, find the largest over-approximation of $\sqrt{11}$ on the interval $[9,16]$.

14. Find the equation of the line parallel to $y=3\ln(2x)$ at $x=-3$ that goes through the point $(2,1)$.

Normal Lines

1. What is the slope of the normal line to the graph $y=x^3$ at the point $(2,8)$?

2. Find the slope of the line normal to $f(x)=x^2\cdot 2^x$ at $x=2$.

3. If $a$ equals the slope of the tangent line to the graph of $y=\sin(\ln(x+1)+5x)$ at $x=0$ and $b$ equals the slope of the normal line to the graph $y$ at $x=0$, then find $a+b$.

4. Find the slope of the normal line to $y=x+\cos{xy}$ at $(0,1)$.

5. Find the slope of the line normal to the parametric curve defined by $x=4t^2-3t$ and $y=6e^t+4$ at $t=\ln4$.

6. Two lines normal to the graph $y=x^2$ are drawn. Denote line $L_1$ as the normal line at $x=1$ and $L_2$ as the normal line at $x=2$. Find the $y$-coordinate of the point of intersection of lines $L_1$ and $L_2$.

7. Find the equation of the line normal to the equation $2y^2=x(7xy+yx^2-8)$ at $x=1$.

Tabular Derivatives

\begin{array}{|c|c|c|c|c|} \hline x & f(x) & g(x) & f'(x) & g'(x) \\ \hline 3&-3&6&-5&1\\ \hline 4&0&3&-3&9\\ \hline 5&3&-2&4&5\\ \hline \end{array}
1. The table above shows some of the values of 2 differentiable functions and their derivatives. If $h(x)=f(x)g(x)$, find $h'(5)$.

2. Using the values in the table from the previous problem, if $h(x)=\frac{f(x)}{g(x)}$, find $h'(4)$.

3. \begin{array}{|c|c|c|c|} \hline x&1&2&3\\ \hline f(x)&2&3&4\\ \hline f'(x)&-2&3&5\\ \hline g'(x)&-5&6&4\\ \hline \end{array}
4. Looking at the table above, if $[f(x)g(x)]'=7$ when $x=3$, then what is the value of $f'(3)g(3)$?

5. Using the table from the previous problem, find $[g(f(x))]'$ when $x=1$.

6. Using the table from the previous problem and given that $h(x)=f^{-1}(x)$, find $h'(2)$.

7. \begin{array}{|c|c|c|c|c|} \hline x&f(x)&g(x)&f'(x)&g'(x)\\ \hline 0&3&0&-4&3\\ \hline 1&5&2&-1&-3\\ \hline 2&1&5&6&-2\\ \hline 3&7&9&12&-1\\ \hline \end{array}
8. Referring to the table above, if $h(x)=g(f(x))$, find $h'(2)$.

9. By using the table from the previous problem, if $h(x)=g^{-1}(x)$, find $h'(2)$.

10. \begin{array}{|c|c|c|c|c|} \hline x&f(x)&g(x)&f'(x)&g'(x)\\ \hline 2&0.25&3&2&1\\ \hline 4&3&-2&-5&-3\\ \hline \end{array}
11. Given the data in the chart, let $h(x)=\frac{g^2(x^2)}{(x+2)f(x)}$. Evaluate $h'(2)$.

12. \begin{array}{|c|c|c|} \hline x&f(x)&f'(x)\\ \hline -1&2&3\\ \hline 1&4&1\\ \hline 2&5&2\\ \hline \end{array}
13. $f(g(x^2))=x$. $f(x)$ is a monotonic function. Using the chart, solve for the value of $g'(4)$.