Q: Which of the following is the correct relationship between the hyperbolic tangent and hyperbolic sine and cosine?

$$\tanh(x) = \dfrac{\sinh(x)}{\cosh(x)}$$

Q: Which of the following is the correct formula for the directional derivative of $$f(x, y) at $$the point $$(x_0$, y_0$) in $$the direction of a unit vector $$\vec{u}$$?

$$D_{\vec{u}}f(x_0$, y_0$) = \nabla f(x_0$, y_0$) \cdot \vec{u}$$

Question 1

Which of the following is the correct definition of the hyperbolic sine function $$\sinh(x)$$?

Accepted Answer

$$\sinh(x) = \dfrac{e^x - e$$^{-x}$$}{2}$$

Question 2

Which of the following Taylor series expansions correctly represents $$\cos(x)$$ centered at $$x=0$$?

Accepted Answer

$$\cos(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n x$$^{2n}$$}{(2n)!}$$

Question 3

A chemical reaction's rate is modeled by $$f(t) = e$$^{-0.5t}$$. Use the second-degree Taylor polynomial centered at t=0$ to approximate the rate at t=0.2$. Which is the correct approximation?

Accepted Answer

$$f(0.2) \approx 1 - 0.1 + 0.01 = 0.91$$

Question 4

Which of the following statements about the Taylor series is TRUE?

Accepted Answer

The Taylor series of a function $$f(x)$$ centered at $$a is$$ $$\displaystyle\sum_{n=0}^{\infty} \dfrac{f$$^{(n)}$$(a)}{n!}(x-a)^n$$

Question 5

Which of the following is the correct relationship between the hyperbolic tangent and hyperbolic sine and cosine?

Accepted Answer

$$	anh(x) = \dfrac{\sinh(x)}{\cosh(x)}$$

Question 6

Suppose the temperature at a point $$(x, y, z) in a $$room is given by $$T(x, y, z) = 100 - x^2$ - y^2$ - z^2$. What is the gradient $$
abla T$$ at the point (1$$, 2, 2)$$?

Accepted Answer

$$
abla T = \left$$

Question 7

Which of the following is the equation of a sphere of radius $$r$$ centered at $$(a, b, c) in $$three-dimensional space?

Accepted Answer

$$(x-a)^2$$ + $$(y-b)^2$$ + $$(z-c)^2$$ = $$r^2$$

Question 8

A drone moves in space along the path $$\vec{r}(t) = \left. $$What is the velocity vector at $$t=2$$?

Accepted Answer

$$\left$$

Question 9

Which of the following is the correct formula for the partial derivative of $$f(x, y) = x^2$y + y^3$ with respect to y$?

Accepted Answer

$$\dfrac{\partial f}{\partial y} = x^2$ + 3y^2$

Question 10

A company’s profit $$P(x, y)$$ depends on the number of units $$x$$ and $$y of $$two products sold. If $$P(x, y) = 5x + 3y - x^2$ - y^2$, what is the maximum profit, and at what values of x$ and y$ does it occur?

Accepted Answer

Maximum profit is $$8 at$$ $$x=2.5$$, $$y=1.5$$

Question 11

Which of the following is the correct formula for the directional derivative of $$f(x, y) at $$the point $$(x_0$, y_0$) in $$the direction of a unit vector $$\vec{u}$$?

Accepted Answer

$$D_{\vec{u}}f(x_0$, y_0$) = 
abla f(x_0$, y_0$) \cdot \vec{u}$$

Question 12

Which of the following is the correct Taylor series expansion for $$e^x$$ centered at $$x=0$$?

Accepted Answer

$$e^x = \displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}$$

Question 13

A medical researcher models the spread of a virus in a tissue as $$f(x, y) = e$$^{-(x^2 + y^2)}$$. What is the value of $$\dfrac{\partial f}{\partial x}$$ at the point (1$$, 0)$$?

Accepted Answer

$$\dfrac{\partial f}{\partial x}(1, 0) = -2e$$^{-1}$$

Question 14

Which of the following is the correct formula for the chain rule for a function $$z = f(x, y)$$ where $$x = g(t)$$ and $$y = h(t)$$?

Accepted Answer

$$\dfrac{dz}{dt} = \dfrac{\partial f}{\partial x} \dfrac{dx}{dt} + \dfrac{\partial f}{\partial y} \dfrac{dy}{dt}$$

Question 15

Which of the following is the correct formula for the hyperbolic cosine function $$\cosh(x)$$?

Accepted Answer

$$\cosh(x) = \dfrac{e^x + e$$^{-x}$$}{2}$$

Question 16

Suppose $$f(x, y, z) = xyz. $$What is $$\dfrac{\partial f}{\partial y} at $$the point $$(2, 3, 4)$$?

Accepted Answer

$$\dfrac{\partial f}{\partial y}(2, 3, 4) = 8$$

Question 17

A weather balloon’s position is given by $$\vec{r}(t) = \left. $$What is the speed of the balloon at $$t=1$$?

Accepted Answer

$$\sqrt{9 + 64 + 225} = 18$$

Question 18

Which of the following is the correct formula for the equation of a plane passing through the point $$(x_0$, y_0$, z_0$)$$ with normal vector $$\vec{n} = \left$$?

Accepted Answer

$$a(x - x_0$) + b(y - y_0$) + c(z - z_0$) = 0$$

Calculus Study Guide: Hyperbolic Functions, Taylor Series, Vectors, and Partial Derivatives

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