What is Ap calculus: Definition and 117 Discussions

Advanced Placement Calculus (also known as AP Calculus, AP Calc, or simply AB / BC) is a set of two distinct Advanced Placement calculus courses and exams offered by the American nonprofit organization College Board. AP Calculus AB covers basic introductions to limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus additional topics (including more advanced integration techniques such as integration by parts, Taylor series, parametric equations, vector calculus, polar coordinate functions, and curve interpolations).

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  1. AL107

    Derivatives and the chain rule

    I originally thought you’d have to use the chain rule to get h’, as in: f’(g(x))*g’(x). Plugging in 1 for x, I got an answer of 10. An online solution, however, said that you only had to get f(g(1)), which was f(-1), then look up f’(-1) in the table. Both approaches seem logical to me, but they...
  2. karush

    MHB 3.3.2 AP Calculus Exam interval from f'(x)

    screen shot to avoid typos OK the key said it was D I surfed for about half hour trying to find a solution to this but $f'(0)$ doesn't equal any of these numbers $e^0=\pm 1$ from the $e^{(x^2-1)^2}$ kinda ?
  3. karush

    MHB S8.7.2.1 AP calculus Exam (typo problem)

    ok I thot this was just observation to get b. but maybe not I saw some rather hefty substations to get different answers
  4. karush

    MHB What Is the Value of the Integral $\int_0^1 xe^x \, dx$?

    $\tiny{4.2.5}$ $\displaystyle\int^1_0{xe^x\ dx}$ is equal to $A.\ \ {1}\quad B. \ \ {-1}\quad C. \ \ {2-e}\quad D.\ \ {\dfrac{e^2}{2}}\quad E.\ \ {e-1}$ ok I think this is ok possible typos but curious if this could be solve not using IBP since the only variable is x
  5. karush

    MHB 1.6.1 AP Calculus Exam Limits with L'H

    $\displaystyle\lim_{x \to 0}\dfrac{1-\cos^2(2x)}{(2x)^2}=$ by quick observation it is seen that this will go to $\dfrac{0}{0)}$ so L'H rule becomes the tool to use but first steps were illusive the calculator returned 1 for the Limit
  6. karush

    MHB Q1 Can you pass this 3 question AP calculus Quiz in 10 minutes

    1. $f(x)=(2x+1)^3$ and let g be the inverse function of f. Given that$f(0)=1$ what is the value of $g'(1)$? A $-\dfrac{2}{27}$ B $\dfrac{1}{54}$ C $\dfrac{1}{27}$ D $\dfrac{1}{6}$ E 6 2. given that $\left[f(x)=x-2,\quad g(x)=\dfrac{x}{x^2+1}\right]$ find $f(g(-2))$...
  7. karush

    MHB 2.4.3 AP Calculus Exam Integration limits

    by observation I choose (c) since the limit values may not be =
  8. karush

    MHB What Does the Intermediate Value Theorem Guarantee for a Continuous Function?

    I thot I posted this before but couldn't find it ... if so apologize Let f be a function that is continuous on the closed interval [2,4] with $f(2)=10$ and $f(4)=20$ Which of the following is guaranteed by the $\textbf{Intermediate Value Theorem?}$ $a.\quad f(x)=13 \textit{ has a least one...
  9. karush

    MHB 4.1.1 AP calculus Exam Int with U substitution

    Evaluate $\displaystyle\int{\dfrac{{(1-\ln{t})}^2}{t} dt=}$ $a\quad {-\dfrac{1}{3}{(1-\ln{t})}^3+C} \\$ $b\quad {\ln{t}-2\ln{t^2} +\ln{t^3} +C} \\$ $c\quad {-2(1-\ln{t})+C} \\$ $d\quad {\ln{t}-\ln{t^2}+\dfrac{(\ln{t^3})}{3}+C} \\$ $e\quad {-\dfrac{(1-\ln{t^3})}{3}+C}$ ok we can either expand...
  10. karush

    MHB 4.1.306 AP Calculus Exam Area under Curve

    $\textsf{What is the area of the region in the first quadrant bounded by the graph of}$ $$y=e^{x/2} \textit{ and the line } x=2$$ a. 2e-2 b. 2e c. $\dfrac{e}{2}-1$ d. $\dfrac{e-1}{2}$ e. e-1Integrate $\displaystyle \int e^{x/2}=2e^{x/2}$ take the limits...
  11. karush

    MHB What is the AP calculus exam IVT question?

    Ok I thot I posted this before but after a major hunt no find Was ? With these options since if f(x) Is a curve going below the x-axis Is possible and () vs [] If this is a duplicate post What is the link.. I normally bookmark these Mahalo ahead
  12. karush

    MHB 6.1.1 AP Calculus Inverse of e^x

    If $f^{-1}(x)$ is the inverse of $f(x)=e^{2x}$, then $f^{-1}(x)=$$a. \ln\dfrac{2}{x}$ $b. \ln \dfrac{x}{2}$ $c. \dfrac{1}{2}\ln x$ $d. \sqrt{\ln x}$ $e. \ln(2-x)$ ok, it looks slam dunk but also kinda ? my initial step was $y=e^x$ inverse $\displaystyle x=e^y$ isolate $\ln{x} = y$ the...
  13. karush

    MHB 4.5.1 AP Calculus Exam .... area of piece wise funcion

    ok well it isn't just adding the areas of 2 functions but is $xf(x)$ as an integrand Yahoo had an answer to this but its never in Latex so I couldn't understand how they got $\dfrac{7}{2}$
  14. karush

    MHB 2.7.2 AP Calculus Exam .... distance traveled estimate

    https://www.physicsforums.com/attachments/9542 ok all I did was count squares so about 21
  15. karush

    MHB 1.8.1 AP Calculus find k for critical point

    $\displaystyle g'=2xe^{kx}+e^{kx}kx^2$ we are given $ x=\dfrac{2}{3}$ then $\displaystyle g'=\dfrac{4}{3}e^\left(\dfrac{2k}{3}\right)+e^\left({\dfrac{2k}{3}}\right)\dfrac{4k}{9}$ ok something is ? aren't dx supposed to set this to 0 to find the critical point did a desmos look like k=-3 but ...
  16. karush

    MHB 4.2.251 AP Calculus Exam......must be true on the interval 0<x<2

    https://www.physicsforums.com/attachments/9527 ok from online computer I got this $\displaystyle\int_0^x e^{-t^2}=\frac{\sqrt{\pi }}{2}\text{erf}\left(t\right)+C$ not sure what erf(t) means
  17. karush

    MHB 1.4.293 AP Calculus Exam a(t) and v(t) t=8

    OK, this can only be done by observation so since we have v(t) I chose e but the eq should have a minus sign. here the WIP version of the AP Calculus Exam PDF as created in Overleaf https://documentcloud.adobe.com/link/track?uri=urn%3Aaaid%3Ascds%3AUS%3A053a75d8-ca5b-4447-bd65-4e580f0de793...
  18. karush

    MHB 4.1.310 AP calculus Exam Area under to functions

    ok I got stuck real soon... .a find where the functions meet $$\ln x = 5-x$$ e both sides $$x=e^{5-x}$$ok how do you isolate x? W|A returned $x \approx 3.69344135896065...$ but not sure how they got itb.? c.?
  19. karush

    MHB 2.6.284 AP calculus Exam Lamp and Shadow related rates

    yes I know this is a very common problem but likewise many ways to solve it ok I really have a hard time with these took me 2 hours to do this looked at some examples but some had 3 variables and 10 steps confusing to get the ratios set up... ok my take on it is here see if you can solve...
  20. karush

    MHB 1.6.365 AP Calculus Exam Limits

    ok I chose c being false, since a limit does not exist if f(x) is different coming from $\pm$
  21. karush

    MHB 2.3.361 AP Calculus Exam of differentials of sin wave

    image due to graph, I tried to duplicate this sin wave on desmos but was not able to. so with sin and cos it just switches to back and forth for the derivatives so thot a this could be done just by observation but doesn't the graph move by the transformations well anyway?
  22. karush

    MHB 4.2.251 AP calculus exam concave ?

    ok this one baffled me a little but isn't $g = e^{-t^2}$and the graph of that has an inflection point at x=1
  23. karush

    MHB 2.1.2 AP calculus Exan particle move along the x-axis

    A particle moves along the x-axis. The velocity of the particle at time t is $6t - t^2$. What is the total distance traveled by the particle from time $t = 0$ to $t = 3$ ok we are given $v(t)$ so we do not have to derive it from a(t) since the initial $t=0$ we just plug in the $t=3$ into $v(t)$...
  24. karush

    MHB 3.1.6 AP calculus Exam piece wise integral

    I tried to do this just by observation, but kinda hard with a piece wise function so would presume
  25. karush

    MHB 4.1.286 AP Calculus Exam .... table of f(t).

    ok these always baffle me because f(t) is not known. however if $f'(t)>0$ then that means the slope is aways positive which could be just a line. but could not picture this to work in the tables. Im sure the answer can be found quickly online but I don't learn by copy and paste. d was...
  26. karush

    MHB 1.8.4 AP Calculus Exam Integral of piece-wise function

    image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes
  27. karush

    MHB 3.3.300 AP Calculus f' interval

    image due to macros in overleaf well apparently all we can do is solve this by observation which would be the slope as x moves in the positive direction e appears to be the only interval where the slope is always increasing
  28. karush

    MHB 2.2.1 AP Calculus Exam .... derivative with ln

    If $f(x)=7x-3+\ln(x),$ then $f'(1)=$ $a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$ see if you can solve this before see the proposed solution
  29. karush

    MHB 303 AP calculus int with initial value

    ok image to avoid typo... try to solve before looking at suggested solutions ok I think you could do this by observation if you are careful with signs
  30. karush

    MHB AP calculus exam tikx graph of e (tan x ) -2

    The graph of $y=e^{\tan x} - 2$ crosses the x-axis at one point in the interval [0,1]. What is the slope of the graph at this point. A. 0.606 B 2 C 2.242 D 2.961 E 3.747ok i tried to do a simple graph of y= with tikx but after an hour trying failed doing this in demos it seens the answer is...
  31. karush

    MHB 3.1.5 AP Calculus Exam average temperature

    309 average temperature $$\begin{array}{|c|c|c|c|c|c|c|} \hline t\,(minutes)&0&4&9&15&20\\ \hline W(t)\,(degrees Farrenheit)&55.0&57.1&61.8&67.9&71.0\\ \hline \end{array}$$ The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W. where...
  32. karush

    MHB 2.7.3 AP calculus Exam Riemann sum

    ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate. other wise it is just adding up the 4 $(t)\cdot(R(t))$s.
  33. karush

    MHB 9.1.317 AP calculus exam multiple choice derivatives of sin wave

    ok just posted an image due to macros in the overleaf doc this of course looks like a sin or cos wave and flips back and forth by taking derivatives looks like a period of 12 and an amplitude of 3 so... but to start I was not able to duplicate this on desmos altho I think by observation alone...
  34. karush

    MHB 2.1.314 AP Calculus Exam a particle moves along the x-axis......

    ok again I used an image since there are macros and image I know this is a very common problem in calculus but think most still stumble over it inserted the graph of v(t) and v'(t) and think for v'(t) when the graph is below the x-axis that participle is moving to the left the integral has a...
  35. karush

    MHB 4.2.8 AP calculus Exam Integration limits

    ok I posted a image to avoid any typos but was wondering why the question has dx and options are in dt I picked C from observation but again that was assuming f was a horizontal line of which it could be something else that way the limits stay the same but the area is cut in halfopinions...
  36. karush

    MHB 5.1.313 AP Calculus Exam DE on bird weight

    I just posted a image due to overleaf newcommands and graph ok (a) if we use f(20) then the $B=0$ so their no weight gain. (b), (c), was a little baffled and not sure how this graph was derived...
  37. karush

    MHB 2.1.312 AP Calculus Exam Int of half circle

    The function f is defined by $$f(x)=\sqrt{25-x^2},\quad -5\le x \le 5$$ (a) Find $f'(x)$ apply chain rule $$ \dfrac{d}{dx}(25-x^2)^{1/2} =\dfrac{1}{2}(25-x^2)^{-1/2}2x =-\frac{x}{\sqrt{25-x^2}}$$ (b) Write an equation for the tangent line to the graph of f at $x=-3$...
  38. karush

    MHB 9.2.2 AP Calculus Exam -- slope field for which DE

    ok probably if one did a lot of these this could be solved by observation others separate the variables and take to Integral to get the equation
  39. karush

    MHB 9.2.2 AP Calculus Exam Slope Fields

    I'm just going to post this image now since my tablet won't render the latex. This is a free response question.. But my experience is that the methods of solving are more focused here at mhb saving many error prone steps.. Mahalo ahead...
  40. karush

    MHB 4.2.5 AP Calculus Exam int of e

    calculator returned this but know sure why $\displaystyle2 \int _1^2e^udu$ note there might be a duplicat of this post ?
  41. karush

    MHB 3.3.04 AP Calculus Exam 2nd derivative

    Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$offhand don't know the correct answer $\tiny{from College Board}$
  42. karush

    MHB 3.3.291 AP Calculus Exam Problem solve for k

    Let $g(x)$ be the function given by $g(x) = x^2e^{kx}$ , where k is a constant. For what value of k does g have a critical point at $x=\dfrac{2}{3}$? $$(A)\quad {-3} \quad (B)\quad -\dfrac{3}{2} \quad (C)\quad -\dfrac{3}{2} \quad (D)\quad {0} \quad (E)\text{ There is no such k}$$ ok I...
  43. karush

    MHB 3.4.238 AP calculus exam Limits with ln

    238 Solve $$\displaystyle\lim_{h\to 0} \dfrac{\ln{(4+h)}-\ln{h}}{h}$$ $$(A)\,0\quad (B)\, \dfrac{1}{4}\quad (C)\, 1\quad (D)\, e\quad (E)\, DNE$$ The Limit diverges so the Limit Does Not Exist (E)ok the only way I saw that it diverges is by plotting not sure what the rule is that observation...
  44. karush

    MHB 4.2.236 AP calculus Exam integral with u substitution

    AP Calculas Exam Problem$\textsf{Using $\displaystyle u=\sqrt{x}, \quad \int_1^4\dfrac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$ is equal to which of the following}$ $$ (A)2\int_1^{16} e^u \, du\quad (B)2\int_1^{4} e^u \, du\quad (C) 2\int_1^{2} e^u \, du\quad (D) \dfrac{1}{2}\int_1^{2} e^u \, du\quad...
  45. karush

    MHB 4.1.237 AP calculus exam find area

    $\tiny{237}$ $\textsf{The total area of the region bounded by the graph of $f(x)=x\sqrt{1-x^2}$ and the x-axis is}$ $$(A) \dfrac{1}{3}\quad (B) \dfrac{1}{3}\sqrt{2}\quad (C) \dfrac{1}{2}\quad (D) \dfrac{2}{3}\quad (E) 1 $$ find the limits of integration if $$f(x)=0 \textit{...
  46. karush

    MHB 219 AP Calculus Exam Inverse function

    Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$$(A)\, \dfrac{2}{27} \quad (B)\, \dfrac{1}{54} \quad (C)\, \dfrac{1}{27} \quad (D)\, \dfrac{1}{6} \quad (E)\, 6$ok not sure what the best steps on this would be but assume we first find...
  47. karush

    MHB 2.8.218 AP Calculus Exam f'(g(3))=

    image to avoid typos