Resources

Resources

Find equations of tangent lines using limits

Use logarithmic differentiation to find the derivative of

y

with respect to the independent variable.

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\mathrm{y}=(\cos \mathrm{x})^{\mathrm{x}}

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\cos x-x \tan x

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\ln x(\cos x)^{x-1}

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(\cos x)^{x}(\ln \cos x+x \cot x)

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(\cos x)^{x}(\ln \cos x-x \tan x)

249

. Find the equation of the normal to the cury

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\begin{array}{l} \frac{d y}{d x}=1.84 e^{0.5 x} \text { at } x=1 \\ \frac{0.5 x}{d x} \frac{d y}{d(1)}=3.08 \end{array}

5

. The equation of the tangent to the curve

y=k x+\frac{6}{x}

(-2,-19)

p x+q y=c

. Find the values of the integers

k, p, q

c

c>0

For the function below, find a) the critical numbers; b) the open intervals where the function is increasing; and c) the open intervals where it is decreasing.

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f(x)=\frac{x+7}{x+2}

\begin{aligned} 4y -3x&= 40 \ 4y &= 3x - 30 \end{aligned}

Find the equation of the normal to the curve

y=\frac{x-2}{2 x+1}

at the point where the curve cuts the

x

Given an equation

6 + 4x - 2x^2 = 0

6 + 4x - 2x^2

a - (x + b)^2

A curve in the plane is defined parametrically by the equations

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x=t^{3}+t

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y=t^{4}+2t^{2}

. An equation of the line tangent to the curve at

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t=1

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\text{(A)}\ y=2x

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\text{(B)}\ y=8x

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\text{(C)}\ y=2x-1

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\text{(D)}\ y=4x-5

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\text{(E)}\ y=8x+13

\begin{aligned} & 5x-4y = -10 \ & y=2x-5 \end{aligned}

\begin{aligned} y &\leq \dfrac{1}{3}x-3 \ y &\leq -2x-2 \end{aligned}

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ASK YOUR TEACHER

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PRACTICE ANOTHER

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Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is a finite set, enter your answer using set notation.)

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\sum_{n=0}^{\infty}(2 n) \cdot\left(\frac{x}{8}\right)^{n}

A right triangle has side lengths

7,24

25

as shown below.

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Use these lengths to find

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\tan Y

\cos Y

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\sin Y

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\begin{array}{l} \tan Y= \ \cos Y= \ \sin Y= \end{array}

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Sine, cosine, and tangent ratios: Numbers for side lengths

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A right triangle has side lengths

5, 12 ,

13

as shown below.

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Use these lengths to find

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\tan A

\sin A

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\cos A

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\begin{aligned} &\tan A=\ &\sin A=\ &\cos A= \end{aligned}

Quadratic and Polynomial Functions

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Using a given zero to write a polynomial as a product of line

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Your answer Is Incorrect.

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For the polynomial below,

3

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h(x)=x^{3}+3x^{2}-16x-6

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h(x)

as a product of linear factors.

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h(x)=\boxed{\phantom{answer}}

For the function

f(x)=x^{2}-10

, find the slope of the secant line between

x=-1

x=2

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For the function

f(x)=x^{2}-10

, find the slope of the secant line between

x=-6

x=1

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The exponential function

f

is graphed in the

x y

x

1, y

increases by a factor of

3

. Which of the following could be

f

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1

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f(x)=\left(\frac{1}{3}\right)^{x}

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f(x)=\left(\frac{1}{3}\right)^{x}+3

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f(x)=3^{x}+2

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f(x)=2(3)^{x}

Consider the curve given by the equation

3 x^{2}+y^{4}+6 x=253

. It can be shown that

\frac{d y}{d x}=\frac{-6(x+1)}{4 y^{3}}.

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Write the equation of the horizontal line that is tangent to the curve and is above the

x

What is the area of the region between the graphs of

f(x)=\sqrt{x+1}

g(x)=2 x-4

x=0

x=3

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1

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\frac{23}{3}

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\frac{5}{3}

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\frac{14}{3}

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-3

What is the area of the region between the graphs of

f(x)=x^{2}+12 x

g(x)=3 x^{2}+10

x=1

x=4

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1

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77

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\frac{64}{3}

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18

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45

The graphs of the functions

f(x)=\sin (x)

g(x)=\frac{1}{2}

2

points on the interval

0<x<\pi

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What is the area of the region bound by the graphs of

f(x)

g(x)

between those points of intersection ?

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1

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\frac{\pi}{3}

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\frac{\pi}{2}

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2-\frac{\pi}{2}

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\sqrt{3}-\frac{\pi}{3}