Refer to Figures $8,9,$ and $10 .$ In each case, choose another point on the tangent line to determine the slope of the curve at $P$.

Kaylee M.

Numerade Educator

Kaylee M.

Numerade Educator

Kaylee M.

Numerade Educator

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.

$$f(x)=x^{2}+3 \quad P(1,4)$$

Kaylee M.

Numerade Educator

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.

$$f(x)=-2 x^{2}+3 x+3$$$\quad$$$P(2,1)$$

Kaylee M.

Numerade Educator

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.

$$f(x)=\sqrt{x+1} \quad P(3,2)$$

Kaylee M.

Numerade Educator

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.

$$f(x)=x^{3} \quad P(2,8)$$

Kaylee M.

Numerade Educator

a) sketch the graph of the given function, and then draw the tangent line at the point $P$. (b) Using your sketch, approximate the slope of the curve at $P,$ (c) Use (1) to determine the exact value of the slope at $P$.

$$f(x)=-x^{2}+2 x-1$$$\quad$$$P(1,0)$$

Kaylee M.

Numerade Educator

Given the curve whose equation is $f(x)=x^{2}+3 .$ Let $P$ be the point (1,4)

(a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 1.01 (ii) 1.001 (iii) 1.0001 (iv) 0.99 (v) 0.999 (vi) 0.9999.

(b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.

Numerade Educator

Given the curve whose equation is $f(x)=\sqrt{x+4} .$ Let $P$ be the point(5,3). (a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 5.01 (ii) 5.001 (iii) 5.0001 (iv) 4.99 (v) 4.999 (vi) 4.9999. (b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.

Numerade Educator

Given the curve whose equation is $f(x)=x^{0.3} .$ Let $P$ be the point (1,1). (a) Determine the slope of the secant line joining $P$ to $Q,$ if $Q$ has as its $x$ -coordinate: (i) 1.001 (ii) 1.00001 (iii) 0.999 (iv) 0.9999. (b) What limiting value does the slope of the secant line appear to be approaching as $Q$ approaches $P ?$

Kaylee M.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$y=x^{2}$ at the point (3,9).

Majid B.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$f(x)=3-2 x-x^{2}$ at the point (-1,4).

Majid B.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$f(x)$ as defined in Exercise 4.

Kaylee M.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$f(x)$ as defined in Exercise 5.

Kaylee M.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$f(x)$ as defined in Exercise 6.

Wendi Z.

Numerade Educator

Determine the derivative at the given point on the curve using equation (2).

$f(x)$ as defined in Exercise 7.

Kaylee M.

Numerade Educator

Find $f^{\prime}(x)$.

(a) $f(x)=53$ (b) Give a geometric explanation for your result.

Majid B.

Numerade Educator

Short segments of the tangent lines are given at various points along a curve. Use this information to sketch the curve.

See Figure 11.

Kaylee M.

Numerade Educator

Short segments of the tangent lines are given at various points along a curve. Use this information to sketch the curve.

See Figure 12.

Kaylee M.

Numerade Educator

Given $f(x)=3 x^{2}-12 x+5 .$ At which point will the curve have slope

(a) $0 ;$ (b) $6 ;$ (c) $-6 ?$

Majid B.

Numerade Educator

Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horizontal; (b) have slope $15 ;$ (c) have slope $36 ?$

Majid B.

Numerade Educator

Given $f(x)=x^{3}-12 x .$ At which points will its tangent line (a) be horizontal; (b) have slope 15; (c) have slope 36?

Majid B.

Numerade Educator

Given $f(x)=\sqrt{x}(\text { a) }$ At which point will the tangent line be vertical? (b) What can you say about the derivative at this point?

Majid B.

Numerade Educator

$$f(x)=\left\{\begin{aligned}4 x-2 & \text { if } x \leq 1 \\x+1 & \text { if } x>1\end{aligned}\right.$$ (a) Sketch the graph of this function. (b) Determine $f^{\prime}(x)$ if $x<1$(c) Determine $f^{\prime}(x)$ if $x>1$ (d) What can you conclude about $f^{\prime}(1) ?$

Kaylee M.

Numerade Educator

$$f(x)=\left\{\begin{array}{l}x^{2} \text { if } x \geq 0 \\x \text { if } x<0\end{array}\right.$$ (a) Sketch the graph of this function. (b) Determine $f^{\prime}(x)$ if $x<0$ (c) Determine $f^{\prime}(x)$ if $x>0$ (d) What can you say about $f^{\prime}(0) ?$

Kaylee M.

Numerade Educator

Given $f(x)=|2 x-5|$ (a) At what point is the function not differentiable? (b) What is the derivative to the left of this point? (c) What is the derivative to the right of this point?

Kaylee M.

Numerade Educator

$f(x)=\{\begin{array}{c}9 x+5 \text { if } x \geq 1 \\ x^{2}+7 x+6 \text { if } x<1\end{array} \text { (a) What is } f^{\prime}(x) \text { if } x>1 ?$ (b) What is \right. $f^{\prime}(x)$ if $x<1 ?(\mathrm{c})$ What is the slope of the curve just to the left of $x=1 ?$ (d) What is $f^{\prime}(1) ?$

Kaylee M.

Numerade Educator

Find the point on the curve $y=x^{3}$ at which the tangent line at (2,8) crosses the curve. (You may want to use the result of Exercise $17 .$ )

Wendi Z.

Numerade Educator

Consider the two functions: $f(x)=x^{1 / 3}$ and $g(x)=x^{4 / 3}$ near $x=0 .$ Using $h=-0.1,-.0 .01,-0.001$ and $-0.0001,(\text { as } h \text { approaches } 0$ from the left), and $h=0.1, .0 .01,0.001$ and 0.0001 (as $h$ approaches 0 from the right). Find the slope of the secant lines passing through $P(0,0)$ and $Q(h, f h)$ ). Does $m_{\tan }(x)$ exist at (0,0)$?$ Why not? (b) Now repeat the process for $g(x) .$ What is the difference in the behavior at $P(0,0)$ for the two functions?

Wendi Z.

Numerade Educator

Suppose that, in the development of the definition of the derivative, we wrote $(x_{2}, f(x_{2}) \text { for } Q \text { instead of }(x+h, f(x+h)) .$ Show that the definition of. the derivative will then have the following alternate form: $f^{\prime}(x)=\lim _{x_{2} \rightarrow x} \frac{f\left(x_{2}\right)-f(x)}{x_{2}-x}$

Wendi Z.

Numerade Educator

Use the alternate form of the derivative given in Exercise $37,$ to compute $f^{\prime}(x)$ for the function defined in: (a) Exercise 21; (b) Exercise 22; (c) Exercise 23; (d) Exercise 24.

Wendi Z.

Numerade Educator

Let $y 1(x)=x^{2}+1,$ determine the equation of the secant line through each of the following $x$ -values and $x=2$ :

(a) $x=2.1,$ call the equation $y 2(x)$ and enter it the $Y=$ screen

(b) $x=2.05,$ call the equation $y 3(x)$ and enter it the $Y=$ screen

(c) $x=2.025,$ call the equation $y 4(x)$ and enter it the $Y=$ screen

(d) choose an appropriate window so the curve and all these secant lines can be seen.

(e) Have the calculator add the tangent line at $x=2$

What is happening to the secant lines as $x$ approaches $2 ?$

Wendi Z.

Numerade Educator